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Set CSp(m; R)= (AeGL(2m; R); tAJA=cJ, cat+) =Sp(m; R) x R -E, csp(m; R)= ...... the fibre in aivE -3 N is 2r — 1, the ...

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Shoshichi Kobayashi

Transformation Groups in Differential Geometry Reprint of the 1972 Edition

Springer

Shoshichi Kobayashi Department of Mathematics, University of California

Berkeley, CA 94720-3840 USA

Originally published as Vol. 70 of the Ergebnisse der Mathematik und ihrer Grenzgebiete, 2nd

sequence

Mathematics Subject Classification (1991): Primary 53C20, 53C10, 53C55, 32M05, 32)15, 57S15 Secondary 53C15, 53A10, 53A20, 53A30, 32H20, 58D05

ISBN 3-540-58659-8 Springer-Verlag Berlin Heidelberg New York

Photograph by kind permission of George Bergman

CIP data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provision of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1995 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN 10485278

41/3140 - 5 4 3 2 1 0 - Printed on acid-free paper

Shoshichi Kobayashi

Transformation Groups in Differential Geometry

Springer-Verlag Berlin Heidelberg New York 1972

Shoshichi Kobayashi University of California, Berkeley, California

AMS Subject Classifications (1970): Primary 53 C 20, 53 C 10, 53 C 55, 32 M 05, 32 J 25, 57 E 15 Secondary 53 C 15, 53 A 10, 53 A 20, 53 A 30, 32 H 20, 58 D 05

ISBN 3-540-05848-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-05848-6 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright All rights are reserved, whether the whole or part of the material Is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 0) by Springer-Verlag Berlin Heidelberg 1972. Library of Congress Catalog Card Number 72-80361. Printed in Germany. Printing and binding: Universitatsdruckerei H. Startz AG, Wfirzburg

Preface

Given a mathematical structure, one of the basic associated mathematical objects is its automorphism group. The object of this book is to give a biased account of automorphism groups of differential geometric structures. All geometric structures are not created equal; some are creations of gods while others are products of lesser human minds. Amongst the former, Riemannian and complex structures stand out for their beauty and wealth. A major portion of this book is therefore devoted to these two structures. Chapter I describes a general theory of automorphisms of geometric structures with emphasis on the question of when the automorphism group can be given a Lie group structure. Basic theorems in this regard are presented in §§ 3, 4 and 5. The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geometric structures in a unified manner. In § 8, we sketch the relationship between the two concepts. Chapter I is so arranged that the reader who is primarily interested in Riemannian, complex, conformal and projective structures can skip §§ 5, 6, 7 and 8. This chapter is partly based on lectures I gave in Tokyo and Berkeley in 1965. Contents of Chapters II and III should be fairly clear from the section headings. It should be pointed out that the results in §§ 3 and 4 of Chapter II will not be used elsewhere in this book and those of §§ 5 and 6 of Chapter II will be needed only in §§ 10 and 12 of Chapter III. I lectured on Chapter II in Berkeley in 1968; Chapter II is a faithful version of the actual lectures. Chapter IV is concerned with automorphisms of affine, projective and conformal connections. We treat both the projective and the conformal cases in a unified manner. Throughout the book, we use Foundations of Differential Geometry as our standard reference. Some of the referential results which cannot be found there are assembled in Appendices for the convenience of the reader. As its 'title indicates, this book is concerned with the differential geometric aspect rather than the differential topological or homological

Preface

VI

aspect of the theory of transformation groups. We have confined ourselves to presenting only basic results, avoiding difficult theorems. To compensate for the omission of many interesting but difficult results, we have supplied the reader with an extensive list of references. We have not touched upon homogeneous spaces, partly because they form an independent discipline of their own. While we are interested in automorphisms of given geometric structures, the differential geometry of homogeneous spaces is primarily concerned with geometric objects which are invariant under given transitive transformation groups. For the convenience of the reader, the Bibliography includes papers on the geometry of homogeneous spaces which are related to the topics discussed here. In concluding this preface, I would like to express my appreciation to a number of mathematicians: Professors Yano and Lichnerowicz, who interested me in this subject through their lectures, books and papers; Professor. Ehresmann, who taught me jets, prolongations and infinite pseudo-groups; K. Nomizu, T. Nagano and T. Ochiai, my friends and collaborators in many papers; Professor Matsushima, whose recent monograph on holomorphic vector fields influenced greatly the presentation of Chapter III; Professor Howard, who kindly made his manuscript on holomorphic vector fields available to me. I would like to thank Professor Remmert and Dr. Peters for inviting me to write this book and for their patience. I am grateful also to the National Science Foundation for its unfailing support given to me during the preparation of this book. January, 1972

S. Kobayashi

Contents

1 I. Automorphisms of G-Structures 1 1. G-Structures 5 2. Examples of G-Structures 3. Two Theorems on Differentiable Transformation Groups. • 13 16 4. Automorphisms of Compact Elliptic Structures 19 5. Prolongations of G-Structures 23 6. Volume Elements and Symplectic Structures 28 7. Contact Structures 8. Pseudogroup Structures, G-Structures and Filtered Lie Alge33 bras 39 II. Isometries of Riemannian Manifolds 1. The Group of Isometries of a Riemannian Manifold. . . 39 2. Infinitesimal Isometries and Infinitesimal Affine Trans42 formations 3. Riemannian Manifolds with Large Group of Isometries . 46 55 4. Riemannian Manifolds with Little Isometries 59 5. Fixed Points of Isometries • 67 6. Infinitesimal Isometries and Characteristic Numbers . . 77 III. Automorphisms of Complex Manifolds 1. The Group of Automorphisms of a Complex Manifold . . 77 2. Compact Complex Manifolds with Finite Automorphism 82 Groups 3. Holomorphic Vector Fields and Holomorphic 1-Forms . • 90 4. Holomorphic Vector Fields on Kahler Manifolds . . . • 92 95 5. Compact Einstein-Kahler Manifolds 6. Compact Kahler Manifolds with Constant Scalar Curvature 97 100 7. Conformal Changes of the Laplacian 8. Compact Kahler Manifolds with Nonpositive First Chern Class 103

Contents

VIII

9. Projectively Induced Holomorphic Transformations. . . . 106 10. Zeros of Infinitesimal Isometries 112 115 11. Zeros of Holomorphic Vector Fields 12. Holomorphic Vector Fields and Characteristic Numbers. 119 IV. Affine, Conformal and Projective Transformations

122

1. The Group of Affine Transformations of an Affinely Connected Manifold 122 2. Affine Transformations of Riemannian Manifolds 125 3. Cartan Connections 127 4. Projective and Conformal Connections 131 5. Frames of Second Order 139 6. Projective and Conformal Structures 141 7. Projective and Conformal Equivalences 145 Appendices 1. Reductions of 1-Forms and Closed 2-Forms 2. Some Integral Formulas 3. Laplacians in Local Coordinates 4. A Remark on d'd"-Cohomology

150 150 154 157 159

Bibliography

160

Index

181

I. Automorphisms of G-Structures

1. G-Structures

Let M be a differentiable manifold of dimension n and L(M) the bundle of linear frames over M. Then L(M) is a principal fibre bundle over M with group GL(n; R). Let G be a Lie subgroup of GL(n; R). By a G-structure on M we shall mean a differentiable subbundle P of L(M) with structure group G. There are very few general theorems on G-structures. But we can ask a number of interesting questions on G-structures, and they are often very difficult even for some specific G. It is therefore essential for the study of G-structures to have familiarity with a number of examples. In general, when M and G are given, there may or may not exist a G-structure on M. If G is a closed subgroup of GL(n; R), the existence problem becomes the problem of finding cross sections in a certain bundle. Since GL(n; R) acts on L(M) on the right, a subgroup G also acts on L(M). If G is a closed subgroup of GL(n; R), then the quotient space L(M)/G is the bundle with fibre GL(n; R)/G associated with the principal bundle L(M). It is then classical that the G-structures on M are in a natural one-to-one correspondence with the cross sections

M L(M)/G (see, for example, Kobayashi-Nomizu [1, vol. 1; pp. 57-58]). The socalled obstruction theory gives necessary algebraic-topological conditions on M for the existence of a G-structure (see, for example, Steenrod [1]). A G-structure P on M is said to be integrable if every point of M has a coordinate neighborhood U with local coordinate system x', , xn such that the cross section (a/3x1 , Olaf) of L(M) over U is a cross section of P over U. We shall call such a local coordinate system x1 , , xn admissible with respect to the given G-structure P. If x', , x" and y', , y" are two admissible local coordinate system in open sets U and 1/ respectively, then the Jacobian matrix (a yi/axi)i, is in G at each point of U n V

2

L Automorphisms of G-Structures

Proposition 1.1. Let K be a tensor over the vector space R" (j. e., an element of the tensor algebra over R") and G the group of linear transformations 01W leaving K invariant. Let P be a G-structure on M and K the tensor field on M defined by K and P in a natural manner (see the proof below). Then P is integrable if and only if each point of M has a coordinate neighborhood with local coordinate system x', x" with respect to which the components of K are constant functions on U. Proof We give the definition of K although it is more or less obvious. At each point x of M, we choose a frame u belonging to P. Since u is a linear isomorphism of R" onto the tangent space T(M), it induces an isomorphism of the tensor algebra over Rn onto the tensor algebra over T(M). Then Kx is the image of K under this isomorphism. The invariance of K by G implies that Kx is defined independent of the choice of u. Assume that P is integrable and let x', , f be an admissible local

coordinate system. From the construction above, it is clear that the components of K with respect to x', , f coincide with the components of K with respect to the natural basis in R" and, hence, are constant functions. Conversely, let x1 , , f be a local coordinate system with respect to which K has constant components. In general, this coordinate system is not admissible. Consider the frame (/ax', a/axn) at the origin of this coordinate system. By a linear change of this coordinate system, we obtain a new coordinate system y', , y" such that the frame (ô/ay', ..., at the origin belongs to P. Then K has constant components with respect to y', , y". These constant components coincide with the comabayn) ponents of K with respect to the natural basis of R" since Way', at the origin belong to P. Let u be a frame at x e U belonging to P. Since the components of K with respect to u coincide with the components of K with respect to the natural basis of R" and, hence, with the components of K with respect to (a/ ...,alay n), it follows that the frame (ô14', ...,a/ayn) at x coincides with u modulo G and, hence, belongs to P. q e. d.

amyl)

ay',

Proposition 1.2. If a G-structure P on M is integrable, then P admits a

torsionfree connection. U be a coordinate neighborhood with admissible local coordinate system x', , xn. Let cou be the connection form on Pi U defining a fiat affine connection on U such that alax,,...,alaxn are Proof Let

parallel. We cover M by a locally finite family of such open sets U. Taking a partition of unity tfu} subordinate to { U), we define a desired

3

1. G-Structures

connection form co by

a)=En*fu • wu, where

7t:

P M is the projection.

q.e.d.

In some cases, the converse of Proposition 1.2 is true. For such examples, see the next section. Let P and P' be G-structures over M and M'. Letfbe a diffeomorphism of M onto M' and f* : L(M)--* L(M) the induced isomorphism on the bundles of linear frames. If f, maps P into P', we call f an isomorphism of the G-structure P onto the G-structure P. If M=M' and P = P', then an isomorphism f is called an automorphism of the G-structure P. A vector field X on M is called an infinitesimal automorphism of a G-structure P if it generates a local 1-parameter group of automorphisms of P. As in Proposition 1.1, we consider those G-structuresdefined by a tensor K. Then the following proposition is evident. Proposition 1.3. Let K be a tensor over the vector space Rn and G the group of linear transformations of Rn leaving K invariant. Let P be a Gstructure on M and K the tensor field on M defined by K and P. Then (1) A diffeomorphismf: M M is an automorphism of P if and only iff leaves K invariant; (2) A vector field X on M is an infinitesimal automorphism of P if and only if Lx K =0, where Lx denotes the Lie derivation with respect to X. We shall now study the local behavior of an infinitesimal automor-

phism of an integrable G-structure. Without loss of generality, we may assume that M =Rn with natural coordinate system x 1, , xn and P =Rn x G. Let X be a vector field in (a neighborhood of the origin of) Rn and expand its components in power series:



E

1

E

x31 xik,

kr. 0 — • )1, ..., fic .1

where 4 are symmetric in the subscripts j1 , Since X is an infinitesimal automorphism of P if and only if the matrix (a viaxi) belongs to the Lie algebra g of G, we may conclude that X is an infinitesimal automorphism of P if and only if, for each fixed j 2 , ...,jk , the matrix (4 n belongs to the Lie algebra g. This motivates the following definition.

4

1. Automorphisms of G-Structures

Let g be a Lie subalgebra of gl(n; R). For k =0, 1, 2, ..., let gk be the space of symmetric multilinear mappings

t: R'' x - - • x R" R" ,—....,—. (k +1)-times

such that, for each fixed v 1 , ... , vic e R", the linear transformation ye R" -- t(v, v1 , ..., vk)eR" belongs to g. In particular, go = g. We call gk the k-th prolongation of g. The first integer k such that gk =0 is called the order of g. If gk =0, then 9k+1=gk+ 2 = • • • =0. If g,,0 for all k, then g is said to be of infinite type. Proposition 1.4. A Lie algebra g c gl (n; R) is of infinite type if it contains a matrix of rank 1 as an element. Proof Let e be a nonzero element of R" and a a nonzero element of

the dual space of W. Then the linear transformation defined by veR"-- eeR" is of rank 1, and conversely, every linear transformation of rank 1 is given as above. Assume that the transformation above belongs to g. For each positive integer k, we define t(vo , V1 , ... , vk) = - - - e, Then t is a nonzero element of gk .

vi e R".

q.e. d.

We say that a Lie algebra g c gl(n; R) is elliptic if it contains no matrix of rank 1. Proposition 1.4 means that if g is of finite order, then it is elliptic. Each Lie subalgebra g of gl(n; R) gives rise to a graded Lie algebra CO

E 9k, where g_ 1 =Rn. The bracket of te gp and t'Egq is defined by

k = —1

[t, tl (vo, vi, ... , vp+ q)=

1 , , vj), vj,,, ...vi„ , +„) E t(e( vjo... p! (q + I)! 1 (p+ I)! q!

E

c0

,

...

,

vkp ), vk,„ ... , v kp+ ,7 )•

In particular, if tEgp , p. 0, and v e g_ i =R", then

[t, v](v i , ... , vp)=t(v, v 1 , ... , v a). We explicitly set [g_ 1 , g_ 1 ] =0. This definition is motivated by the following geometrical consideration. Suppose t=(4 ... jp )e gp and t' = (b ki o... kci )e gq in terms of components and consider the corresponding

2. Examples of G-Structures

vector fields:

x. Y

5

1 (p+1)!

E dk..-JP xi°

xi

a

*** P axi)

a

1

(q±1)! co

Then [X, Y] corresponds to [t,

Thus, the graded Lie algebra

E gk

may be considered as the Lie algebra of infinitesimal automorphisms

E

.— a with polynomial components c axl

of the flat G-structure P =

R" x G on Rn. For a survey on G-structures, see expository articles of Chern [1], [2]; the latter contains an extensive list of publications on the subject. See also Sternberg's book [1], A. Fujimoto [2], [3], Bernard [1]. The group of automorphisms of a compact elliptic structure or a G-structure of fmite type will be shown to be a Lie transformation group (see §§ 4 and 5, respectively). These two cases cover a substantial number of interesting geometric structures whose automorphism groups are Lie groups. By considering G-structures of higher degree, we can bring such structures as projective structures under this general scheme (see § 8 of this chapter and Chapter IV). The group of automorphisms of a bounded domain or a similar complex manifold is also a Lie group (see § 1 of Chapter III), but this does not come under the general scheme. This book does not touch area-measure structures (Brickell [1]), nor pseudo-conformal structures of real hypersurfaces in complex manifolds (MorimotoNagano [1], Tanaka [3]) although automorphism groups of these structures are usually Lie groups. 2. Examples of G-Structures Example 2.1. G= GL(n; R) and g = gl (n; R). The Lie algebra g contains a matrix of rank 1 and is of infinite type. A G-structure on M is nothing but the bundle L(M) of linear frames and is obviously integrable. Every diffeomorphism of M onto itself is an automorphism of this G-structure and every vector field on M is an infinitesimal automorphism. Example 2.2. G = GL+ (n; R) and g = g! (n; R), where GL+ (n; R) means the group of matrices with positive determinant. The Lie algebra g is of infinite type. A manifold M admits a GL+ (n; R)-structure if and only if it is orientable; this is more or less the definition of orientability. A GL+ (n; R)-structure on M may be considered as an orientation of M and is obviously integrable. A diffeomorphism of M onto itself is an

6

I. Automorphisms of G-Structures

automorphism of a GL+ (n; R)-structure if and only if it is orientation preserving. Every vector field on M is an automorphism since every oneparameter group of transformations is orientation preserving.

Example 2.3. G= SL(n; R) and g _—%1(n; R). Again, g contains a matrix of rank 1 and is of infinite type. The natural action of GL(n; R) on R" induces an action of GL(n; R) on A" Rn such that

A v det (A) • v

for A e GL(n ; R) and v ez1" Rn.

The group GL(n; R) is transitive on A" R"— {0} with isotropy subgroup SL(n; R) so that A" — {0} = GL(n; R)/SL(n; R). It follows that the cross sections of the bundle L(M)/SL(n; R) are in one-to-one correspondence with the volume elements of M, L e., the n-forms on M which vanish nowhere. In other words, an SL(n; R)-structure is nothing but a volume element on M. It is clear that M admits an SL(n; R)-structure if and only if it is orientable. We claim that every SL(n; R)-structure is integrable. Indeed, let U be a coordinate neighborhood with local coordinate system x" and let (p =f dx 1 A • • • A dxn be the volume element corresponding to the given SL(n; R)-structure. Let = y 1 (x', , xn) be a function such that ay 1lax1 =f Then dx" = dy l A dX 2 A

= f dx l

•••

A dx",

which shows that the coordinate system y 1 , x2, x" is admissible with respect to the given SL(n; R)-structure. A diffeomorphism of M onto itself is an automorphism of the SL(n; R)-structure if and only if it preserves the volume element (p. Let X be a vector field on M. The function (5X defined by L x = (6 X) • (p is called the divergence of X with respect to (p. Clearly, X is an infinitesimal automorphism of the SL(n; R)-structure if and only if (5 X =0. For SL(n; R)-structures, see § 6.

Example 2.4. G = GL(m; C) and g = gI(m; C). We consider GL(m; C) (resp. gl (m; C)) as a subgroup of GL(2 m; R) (resp. a subalgebra of gl(2 m; R)) in a natural manner, i.e.,

A A2 ) ' Ai eGL(2m, R) A l +i A 2 EGL(m; C) _ ( _ A2 or gl (2 m ; R) .

or gI(m; C)

Let z1 , , zm be the natural coordinate system in Cm and zi= xi+ i xrn+ j =1, , m. Then the identification e" =R 2 ' given by z1 es) 2 m

,

...

...,

X

)

2. Examples of G-Structures

7

induces the preceding injections GL(m; C) --*GL(2 m; R)

gl(2m; R).

and gl(m;

The multiplication by i in Cm, i.e., , zm)--*

izm),

induces a linear transformation xm, xm+ 1 , ..., X 2 m) .--*( -Xm+1 ,

- X 2 m, X 1 ,

xm)

of R2 m, which will be denoted by J. Since i 2 = —1, we have J 2 = —I. In matrix form, —I J = (O I 01 The group GL(m; C) (resp. the algebra gi(m; C)), considered as a subgroup of GL(2 m; R) (resp. a subalgebra of gl (2 m; R)), is given by GL(m, C)= {AEGL(2m; R); AJ=J A} gl(m; C). {Aeg1(2m; R); AJ =JA}. Since gk consists of all symmetric multilinear mappings of Cm x • • • x Cm (k + 1 times) into Cm, the Lie algebra g is of infinite type. Every element of g, considered as an element of gl(2 m; R) is of even rank. Hence, g is elliptic. The GL(m; C)-structure on a manifold M (of dimension 2m) are in one-to-one correspondence with the tensor field J of type (1, 1) on M such that Jx Jx = — Ix (or simply, J J = — I), where ./x is the endomorphism of the tangent space TX(M) given by J and Ix is the identity transformation of T(M). The correspondence is given as follows. Given a tensor field J with J 0J. I, we consider, at each point x of M, only those linear frames u: R 2 m Tx (M) satisfying u 0 J = Jo The subbundle of L(M) thus obtained is the corresponding GL(m; 0-structure on M. A tensor field J with Jo J = — I or the corresponding GL(m; C)-structure is called an almost complex structure. We claim that an almost complex structure is integrable (as a GL(m; C)structure) if and only if it comes from a complex structure. (Before we explain this statement, we should perhaps remark an almost complex structure J is often called integrable if a certain tensor field of type (1, 2), called the torsion or Nijenhuis tensor, vanishes.) It is a deep result of Newlander and Nirenberg [1] that the two definitions coincide. For the real analytic case, see, for instance, Kobayashi-Nomizu [1, vol. 2; p. 321]. The theorem of Newlander-Nirenberg is equivalent to the statement that an GL(m; C)structure is integrable if and only if it admits a torsionfree affine connec—

8

1. Automorphisms of G-Structures

tion (see Fr6hlicher [1]). Let M be a complex manifold of complex dimension m with local coordinate system z 1, z'n where zi= xi+ i yi. We have the natural almost complex structure J on M defined by J(a/axi)= 3/3y1

j= 1, , m,

J (a/a yi)= —a/axi

j= 1, , m.

The almost complex structure J thus obtained is integrable since (a/ax',

,

gives a local cross section of the GL(m, C)-structure defined by J. Conversely, if an almost complex structure J is integrable as a GL(m; C)structure and if x1 , , x2 m is an admissible local coordinate system, then J(3/3x-1)=alaxm±i and J(a/axm+i)= —a/axi for j= 1, ..., m. If we set zi = xi + i xm+i, then the complex coordinate system z1 , , zm turns M into a complex manifold. A diffeomorphismf of M onto itself is an automorphism of an almost complex structure J if and only if f* 0 J=J of* , where f* : T(M)--* T(M) is the differential off If J is integrable, an automorphism f is nothing but a holomorphic diffeomorphism. A vector field X on M is an infinitesimal automorphism of an almost complex structure J if and only if

[X,JY]=J([X, Y])

for all vector field Yon M.

For further properties of an almost complex structure, see KobayashiNomizu [1; Chapter IX]. Example 2.5. G= 0(n) and g = o(n). The Lie algebra g is of order 1. Let te g l and (4 k) the components of t. By definition, t"ik = tip Since o(n) consists of skew-symmetric matrices, we have t.i k = — tf k . Hence, 4k=tf-tjtjt

j

t j

thus proving ti,, =0. To each Riemannian metric on M, there corresponds the bundle of orthonormal frames over M. This gives a one-to-one correspondence between the Riemannian metrics on M and the 0(n)-structures on M. An 0(n)-structure is integrable if and only if the corresponding Riemannian metric is flat, i. e., it has vanishing curvature. An automorphism of an 0(n)-structure is an isometry of the corresponding Riemannian metric. An infinitesimal automorphism of an 0(n)-structure is an infinitesimal isometry or Killing vector field. We shall discuss isometries and Killing vector fields in detail later (see Chapter II). More generally, let G= 0(p, q), n = p + q, be the orthogonal group ±up2 up2 Then o (p, q) is defined by a quadratic form also of order I. There is a natural one-to-one correspondence between -

9

2. Examples of G-Structures

the pseudo-Riemannian metrics of signature q on M and the O(p, q)structures on M. An 0(p, q)-structure is integrable if and only if the corresponding pseudo-Riemannian metric has vanishing curvature. It should be remarked that, although every paraconipact manifold admits a Riemannian metric, it may not in general admit a pseudo-Riemannian metric of signature q for q #0, n. For automorphism of pseudo-Riemannian manifolds, see Tanno. [1, 2]. Example 2.6. G = CO (n) and g = co (n), n 3. By definition, = c I, cell, c>0},

CO(n)= {ileGL(n; R);

+A=ci,

co(n)=Aegl;R

Thus, CO (n) = 0(n) x R+ and co (n) = o (n) + R, where R+ denotes the multiplicative group of positive real numbers. The Lie algebra co (n) is of order 2 and the first prolongation g1 is naturally isomorphic to the dual space IV* of R. To determine g 1 , let t=(t .k) be an element of g 1 . Since the kernel of the homomorphism A e co(n) --*trace(A)eR is precisely o(n) and since o(n) is of order 1, the linear mapping 1 = (n

t=(ti k)e

Et k

)

eRn*

is injective. The kernel is the first prolongation of o (n). (The factor of 1 — is, of course, not important). To see that this mapping is also surjective, n we have only to observe that = (ic) is the image of t with tj k = 614+ bik To prove g2 =0, let t=(tilik)eg2 . For each fixed k, tl'ik may be considered as the components of an element in g1 and hence can be uniquely written tiljk =4311 jk+ (51.1 ik -45

13k•

Since tIlik must be symmetric in all lower indices, we have E h

from which follows jk = — 6ih• E

From h

E hh =0. From (n —2)

.hh = h

j

h

we obtain (n 2)

fic =

h

from which follows (n —2)

h

h

t— njk = E ?itch,

=E

hh and, hence,

n h

k = — 6Pc' Eh h =0 and n 3, we conclude c J k =0. h

(The reader who prefers an index-free proof is referred to KobayashiNagano [3, III; p. 686].) A CO (n)-structure is called a conformal structure. We say that two Riemannian metrics on M are conformally equivalent if one is a multiple of the other by a positive function. The conformal equivalence classes of Riemannian metrics on M are in a natural one-toone correspondence with the CO(n)-structure on M. A conformal

1. Automorphisms of G-Structures

10

structure is integrable if and only if any Riemannian metric corresponding to the structure is locally conformally equivalent to (dx 1)2 + • • • +(dx")2 respect to a suitable local coordinate system x', , xn. Thus, awith conformal structure is integrable if and only if it is conformally flat in the classical sense (see Eisenhart [1]). Consequently, the integrability of a conformal structure is equivalent to the vanishing of the so-called conformal curvature tensor of Weyl (provided n 3). Given a Riemannian metric g on M, a diffeomorphismf of M onto itself (resp. a vector field X on M) is a conformal transformation, i. e., an automorphism of the conformal structure (resp. an infinitesimal conformal transformation, i. e., an infinitesimal automorphism of the conformal structure) if and only if f * g=p . g (resp. L x g = a • g),

where p (resp. a) is a positive function (resp. a function) on M. Conformal structures and their automorphisms will be discussed in Chapter IV. The reason we excluded the case n =2 is that CO (2) (resp. co (2)) is naturally isomorphic to GL(1; C) (resp. 91(1; C)). For this reason, the conformal differential geometry in dimension 2 differs significantly from that in higher dimensions. In particular, we note that every CO (2)-structure, e., GL(1 ; C)-structure is integrable; this is nothing but the existence of isothermal coordinate systems. The results for CO (n)-structures can be easily generalized to CO (p, q)structures, where CO (p, q)= O (p, q) x R+ is defined by a quadratic form of signature q. Example 2.7. G = U(m) and g = u (m). Since u(m) is a subalgebra of 0(2 m) which is of order 1 (cf. Examples 2.4 and 2.5), it is also of order 1. A U(m)-structure on a 2 m-dimensional manifold M is called an almost hermitian structure; it consists of an almost complex structure and a hermitian metric. Since U(m)= GL(m ; C) 0(2 m), a U(m)-structure may be considered as an intersection of a GL(m ; C)-structure and an 0(2 m)structure. A U(m)-structure is integrable if and only if the underlying almost complex structure is integrable (so that M is a complex manifold) and the hermitian metric has vanishing torsion and curvature. A diffeomorphism of M onto itself is an automorphism of a U(m)-structure if and only if it is an automorphism of the underlying GL(m, C)- and 0(2 m)structures. Similarly, for an infinitesimal automorphism. For automorphisms of hermitian manifolds, see Tanno [3]. Example 2.& G= Sp (m; R) and g =%p (m; R). We recall that Sp (m; R) is the group of linear transformations of R 2 rn leaving the form Ul A Um+1 ± •

—Fan A 142m

2. Examples of G-Structures

11

invariant, where u', ... , u'm is the natural coordinate system in 11 2 m. In other words, Sp (m; R)= {A e GL(2 m; R); `AJ A = J} , sp (m; R)= {Aeg1(2m; R); 'AJ +J A=0} , where J=

(0 I — 10 ) '

Since sp(m; R) consists of matrices of the form A=

(Ai

A2 ) tA 113 ii i

A

: 4.1„

EA

A

'WI L1.1 11 2 = 112

and

'A 3 = A3 9

it contains an element of rank 1 and, hence, is of infinite type. The Sp(m, R)-structures on a 2m-dimensional manifold M are in a natural one-to-one correspondence with the 2-forms co on M of maximum rank (j. e., com 40 everywhere). Since both GL(m; C) and Sp (m; R) contain U(m) as a maximal compact subgroup, a manifold M admits an Sp (m; R)-structure if and only if it admits a GL(m; C)-structure. An Sp(m; R)-structure is called an almost symplectic structure or an almost Hamiltonian structure. If an almost symplectic structure is integrable with admissible coordinate system X', ..., x2 m so that co= dx 1 A dxm+ 1 + • •• + dxm A dx2 m,

then d co =O. Conversely (see Appendix 1), if the form w defining an almost symplectic structure is closed, then co =dx 1 A dxm+ 1-+ • • • + dxm A dx 2 m for a suitable local coordinate system .X 1 9 ... 9 X 2 m and the structure is integrable. An integrable almost symplectic structure is called a symplectic structure or a Hamiltonian structure. We observe that if an almost symplectic structure admits a torsionfree affine connection, then it is integrable. For the 2-form co defining an almost symplectic structure is parallel with respect to such a connection and hence is closed. (In calculating dco in terms of a local coordinate system, partial differentiation may be replaced by covariant differentiation when the connection is torsionfree, see for instance Kobayashi-Nomizu [1, vol. 1; p. 149]). A diffeomorphismf of M onto itself is an automorphism of the symplectic structure defined by a 2-form co if and only iff* co = co. Similarly, X is an infinitesimal automorphism if and only if L x co= 0. An (infinitesimal) automorphism of a symplectic structure is called an ( infinitesimal) symplec tic transformation. Set CSp(m; R)= (AeGL(2m; R); tAJA=cJ, cat+) =Sp(m; R) x R -E, csp(m; R)= Pleg1(2m; R); tAJ +JA=cJ, cell} =sp(m; R)+R.

L Automorphisms of G-Structures

12

A CSp (m; R)-structure is called a conformal-symplectic structure. For conformal-symplectic geometry, see Lee [1]. Example 2.9. G = GL (p, q; R) and g =g1(p, q; R), where GL(p, q; R) denotes the group of linear transformations of 1r, n = p + q, which leave the p-dimensional subspace RP defined by u"' • • • = u =0 invariant. In other words, GL(p, q; R)=I( A B ) . AeGL(p; R), CeGL(q; R)} 0 C

R)= 1(A0 B gRA q;

); Aegl(p; R), Cegl(q; R)}, C

where B denotes a matrix with p rows and q columns. Clearly, g contains an element of rank 1 and, hence, is of infinite type. The GL(p, q; R)structures on M are in a natural one-to-one correspondence with the p-dimensional distributions on M, i. e., the fields of p-dimensional subspaces of tangent spaces. A GL(p, q; R)-structure is integrable if and only if there exists a local coordinate system x 1 , , x" such that a/ax l , aiaxP span the corresponding p-dimensional distribution. In other words, a GL(p, q; R)-structure is integrable if and only if the corresponding p-dimensional distribution is involutive, (see Frobenius theorem). An integrable GL(p, q; R)-structure is known as a foliation with p-dimensional leaves. If a GL (p, q ; 10-structure admits a torsionfree affine connection, it is integrable. Indeed, if X and Yare vector fields belonging to the distribution, then the formula [X, /7] =V Y— Vy X (see KobayashiNomizu [1; p. 133]) implies that [X, Y] also belongs to the distribution. Since an automorphism of a GL(p, q; R)-structure on M is a transformation preserving the corresponding p-dimensional distribution, a vector field X on M is an infinitesimal automorphism if and only if, for every vector field Y belonging to the distribution, [X, Y] belongs to the distribution.

Example 2.10. G = GL(p ; R) x GL (q ; R) and g = gl(p; R)+ gl(q; R), p + q =n. In other words,. GL(p; R) x GL(q ; R)=

{(24 0 Bo)

gl(p; R)-F gl(q; R)={(

; AEGL(p; R), BeGL(q; R)},

A BO) ; Aegl(p; R), Begl(q; R)}. 0

Clearly, g contains an element of rank 1 and, hence, is of infinite type. The GL(p; R) x GL(q; R)-structures are in a natural one-to-one correspondence with the set of pairs (S, S'), where S and S' are complementary distributions of dimensions p and q respectively. A GL(p; R) x GL(q; R)-

3. Two Theorems on Differentiable Transformation Groups

13

structure is integrable if and only if the corresponding distributions S and S' are both involutive, that is, there exists a local coordinate system x', , xn such that a/axi , , 0/axP span S and a/axP+ 1 , , alaxn span S'. Example 2.11. G= {1) and g =O. The {1 } -structures on M are in a natural one-to-one correspondence with the fields of linear frames over M. A manifold M is said to be parallelisable if it admits a (1)structure. The automorphism group of a (1)-structure will be studied in the next section (Theorem 3.2). 3. Two Theorems on Differentiable Transformation Groups The theorems in this section will allow us to prove that the automorphism groups of many geometric structures are Lie groups. Theorem 3.1. Let (1/46 be a group of differentiable transformations of a manifold M. Let S be the set of all vector fields X on M which generate global 1-parameter groups (p i = exp t X of transformations of M such that (A0:5. If the set S generates a finite-dimensional Lie algebra of vector fields on M, then is a Lie transformation group and S is the Lie algebra of IA Proof Let g* be the Lie algebra of vector fields on M generated by S.

a

Let be the connected, simply connected Lie group with Lie algebra g*; it is an abstract Lie group and is not a transformation group. For each element X of g*, we denote by et X the 1-parameter subgroup of generated by X while we denote by exp t X the 1-parameter local group of local transformations of M generated by the vector field X. Then the group acts locally on M in the following sense. There exist a neighborhood U of {I} x M in x M and a mappingf: U --FM such that

a

a

f (et X, p) = (exp t X)p

for (et x , p)e U

x M. Lemma 1. Given X, Ye g*, we define Ze g* by e tz =e x ety e- x, i.e., Z= (ad ex) Y If X, Y are in S, so is Z. Proof of Lemma 1. From et z = ex e1

we obtain

(exp t Z) p = (exp X) (exp t Y) (exp — X) p

If X, YeS, then the right hand side is defined for all p and t. Hence, (exp t Z)p is also defined for all p and t. This implies that Z is in S. Lemma 2. S spans g* as a vector space. Proof of Lemma 2. Let 1/ be the vector subspace of g* spanned by S. By Lemma 1, we have (ad es) S S and, hence, (ad es) V c V Since S

1. Automorphisms of G-Structures

14

generates g*, es generates 6. Hence, (ad () V c V. In particular, (ad el') • V = V, which implies [V, V] c V Lemma 3. S = g*. Proof of Lemma 3. Let X1 , . , X,. E S be a basis for g*. Then the mapping ai Xi eg*

xi

er xr e

gives a diffeomorphism of a neighborhood N of 0 in g* onto a neighborhood U of the identity element in ft. Let YE g*. Let be a positive number such that et 1' e U for I tI In(n-1)+1—n=1-(n

(n — 2)+ 1

From Lemma, it follows that Sx =0(n) or Sx =S0(n). We shall show that is transitive on M. If x and y are two points of M which can be joined by a geodesic, let z be the midpoint of this geodesic segment and let Z be the vector tangent to the geodesic at z. Letfbe a transformation belonging to Sz such that f* (Z)= —Z; such an isometry exists since Sz =--0(n) or Sz = SO (n). Clearly, f (x)= y and f(y)=x. If x and y are arbitrary points of M, we join them by a finite number of geodesic segments and apply the construction above to each segment. In this way, we see that there is an element of (6 which sends x into y. Since (6 is transitive on M, we have r=dimS=dimM+dim Sx =n+dim0(n)=-1-n(n+1).

q.e.d.

Theorem 3.2 is due to H.C.Wang [1]. Lemma used above is due to Montgomery and Samelson [1]. In view of Theorem 3.2, it is natural to ask which Riemannian manifolds of dimension n admits a group of isometries of dimension n (n —1)+ 1. Let M be an n-dimensional Riemannian manifold with n#4. Let E• be a closed subgroup of dimension In(n —1)+1 of 3(M). Let 15x be the isotropy subgroup of at x e M. We shall show that is transitive on M. Assume that it is not. Then, for every xeM, the orbit of e• through

3. Riemannian Manifolds with Large Group of Isometries

49

x is of dimension less than n. Hence,

dim 05„.clim —(n-1)=1n(n-1)+1—(n-1)=1(n-1)(n —2)+1. By Lemma for Theorem 2.2, either 6„ =0(n) or 6„=S0(n). Then, as in the proof of Theorem 3.2, we see that t is transitive on M. Thus, M is a homogeneous Riemannian manifold 6/5, where 5 is a compact group of dimension 1(n —1)(n —2) (=dim 6—n). Lemma 1. Let 5 be a connected closed subgroup of SO(n). If n*4, then 5 is isomorphic to either SO(n —1) or the universal covering group of SO (n —1). If n* 4, 7, then 5 is imbedded in SO (n) as a subgroup leaving a 1-dimensional subspace of Rn invariant. If n =7, then either 5 = SO (n —1) leaving a 1-dimensional subspace of Rn invariant or 5 = Spin(7) with the spin representation. Proof of Lemma I. We shall prove only the first statement and indi-

cate a proof for the remainder of Lemma 1. With respect to an invariant Riemannian metric on the homogeneous space SO (n)/5, the group SO (n) acts as a group of isometries. Since SO(n) is simple for n*4, its action on SO (n)/5 is essentially effective. Since dim SO (n)/$3 =n —1 and dim SO(n)=1n(n —1), Theorem 2.1 implies that SO(n) is a maximal dimensional isometry group acting on SO(n)/b and that SO(n)/5 is either a sphere or a real projective space. Under the linear isotropy representation, 5 is mapped onto SO(n — 1). Hence, 5= 50(n-1) or 5= Spin (n —1). The second and third statements tell us how SO (n —1) or Spin(n —1) can be imbedded into SO(n). The second statement is proved in Montgomery-Samelson [1] by a topological method. We indicate an algebraic proof. First, assume that the action of 5 on Rn is reducible with a p-dimensional invariant subspace W. Then it leaves an (n —p)-dimensional orthogonal complement Rn-13 invariant. Hence, dim b_dim 0(p)+dim 0(n —p)=1p(p —1)+1(n —p)(n—p-1). This implies that p =1 or p=n —1. Next, assume that 5 acts irreducibly on W. Then 5 is absolutely irreducible; otherwise, 5 would be a subgroup of U (n/2) of dimension n'. Now the problem is reduced to that of determining the irreducible representations of degree n of o(n —1; C). But this can be easily accomplished by means of the theory of represenq. e. d. tations of semi-simple Lie algebras. Lemma 2. Let 0 be a connected Lie group of dimension in (n —1)+1 and 5 a connected compact subgroup of 0 of dimension 4 (n —1) (n — 2) such that its linear isotropy representation at a point of M = 0/5 leaves a 1-dimensional subspace of the tangent space invariant. Let g =1) +m' + m"

(vector space direct sum)

IL Isometries of Riemannian Manifolds

50

be an (ad 6)-invariant decomposition of the Lie algebra g, where m' and m" are subspaces of dimension 1 and n-1, respectively. Then there are the following three possibilities, provided n >4:

(1)

[1), nti =0, [m', "] =0,

(2)

[1), nil = 0 > Ent% nil = 0 > [nt", re] = 1); [b, nti =0, [m', m"] =m", [m", mil =0 .

(3)

[m", m"] =0;

and [X,Y]=c Y for X em', Ye m", where c is a constant which depends only on X. Proof of Lemma 2. Since the linear isotropy representation of

of the form

6 is

/1 0 \ k0 SO(n-1)/ '

6 leaves m' elementwise fixed, so that [I), nti =0. We shall show that either [m', m"] =0

or [m', m"] =m".

Fix a non-zero element X of m'. Since the kernel of the linear mapping Yem" —* [X, 1 ] e[m', m"] is invariant by ad 6, it must be either 0 so that dim [m', m"] = dim m" =n-1 or the whole space m" so that [m', m"]=0. We assume dim [m', nt"] =n-1. Since I), nt' and nt" have mutually distinct dimensions so that the irreducible representations of 6 on 1), m' and nt" are mutually inequivalent, it follows that the (n —1)dimensional subspace [ni', mil of g =1)-F m' +nt" invariant by 6 must coincide with m". (Here, we used the assumption n >4.) This proves our assertion. We shall show that if [m', nt"]=nt", then

[X,Y]=cY

for X em' and

Ye m",

where c is a constant which depends only on X and not on Y Since X em' is invariant by 6, the linear isomorphism Ye ni" -+ [X, Ile [m', nt"]=m" commutes with the action of 6 on nt". But 6 acting on nt" is nothing but SO(n —1). Hence, this linear isomorphism is a scalar multiple of the identity transformation. We shall show that either [m", ne] =0

or [ni", nt"] =1) .

Choose a unit vector Xi em' and an orthonormal basis X 2 , ..., X„ for nt". Define the constants C . ic (i, j, k =1, ... , n), by [Xi , XI ]=E Ciik X1 mod 1)

(with

Ciiic = — CO .

3. Riemannian Manifolds with Large Group of Isometrics

51.

We have to prove C ik --.0 for 1 -.I .._n and 2 ..j, ic.n. Fix i, j, k. Choose an integer 1, 2.1-n, such that /4 i, j, k. Since n>4, this is possible. Let A be the linear transformation of m' + nt" defined by

A(Xj). — Xi , A(X 1)= —X,, A(X)=X p for p*j,I. Since A belongs to SO(n —1), it is induced by an element a of $3. From (ad a)([Xi ,Xd)= [(ad a) Xi , (ad a) XJ, we obtain the desired relation ciik =0 by comparing the coefficients of Xi on both sides. Thus, [m", In"] ct). Since [m", ITV] is an ideal of b and since b is simple for n >4, we have either [in", nt"] =0 or [m", nt"] = b. Finally, we prove that [nt",

"] =t) implies

[nt', in"] =0.

Let X e m' and Y, ZEM" be nonzero elements such that [Y, 4 *0. Then [X, [Y, Z]]=[[X, Y], Z] +[Y, [X, Z]]=[c: Y, Z]+[Y, c:Z] =2c[Y, Z].

On the other hand, from [in', 1)] =0, we obtain [X, [ Y, Z] ]=0. Hence, c =0. This completes the proof of Lemma 2. We shall now consider the case where $3 = Spin (7). Lemma 3. Let 15 be a connected Lie group of dimension 29 ( =4.- 8 (8 — 1) + 1) and $3 = Spin (7) such that the linear isotropy representation at a point of M =6/6 is the spin representation. Let

g = b + In

(vector space direct sum)

be an (ad 5)-invariant decomposition of the Lie algebra g. 'Then [In, m] =0 .

b = 21+8 =dim in,

the representations of $ on b and in are mutually inequivalent. On the other hand, [nt, In] is an 6-invariant subspace of g and has dimension 28 ( =48(8 —1)) since dim nt =8. Hence, we have [nt, in] =0, [m, In] = nt or [nt, nt] = b. Assume [in, tn] = in. Let r be the radical of g. It is invariant by b. On the other hand, since b is simple, it follows that dim r dim g — dim b = 8. Hence, r =nt or r =0. Since [m, In] =m, in cannot be solvable. Hence, r =0, i.e., g is semi-simple. Since In is an ideal of g, there is a complementary ideal b'. From dim tr =dim t) and from the fact that the representations of $3 on b and In are inequivalent, it follows that the $3invariant subspace b' must coinéide with b. Hence, [I), in] =0. This contradicts the assumption that the linear isotropy representation of 6 is irreducible so that [b, nt] =nt. We have thus excluded the case [in, m] =m. Proof of Lemma 3. Since dim

II. Isometrics of Riemannian Manifolds

52

Assume [m, m] = I). Let a be an ideal of g. Since the representations of 6 on and nt are inequivalent, an 5-invariant subspace of g must be g, h, nt or O. Hence, a must be g, h, nt or OE But neither h nor nt can be an ideal of g. Hence, either a =g or a =O. This shows that g is simple. But there is no simple Lie algebra of dimension 29. We have thus excluded the case [nt, nt] = h. This completes the proof of Lemma 3. In Lemma 2 and 3, the n-dimensional Riemannian manifolds M such that 3(M) contains a closed subgroup of dimension In(n-1)+ 1 have been locally determined for n >4. We shall now consider the global classification. Consider first the case (1) in Lemma 2. Clearly, M is locally symmetric and flat. If M is simply connected, then M =1In =R X IVI-1 and (6 is the direct product of the group of translations on R and the group of proper motions of R". To find a non-simply connected M, we have to look for a discrete subgroup /1 of the group of motions of Rn which commutes with the above 0 elementwise. It is easy to verify that must be generated by a translation of R. In other words, if M is not simply connected, then M = SI. X where 51 denotes a circle. Consider the case (2) in Lemma 2. Clearly, M is locally symmetric and reducible. If M is simply connected, then M =R x M", where M" must be a space of constant curvature by Theorem 3.1. Since [nt", nt"] = M" is non-flat When M is simply connected, 0 is the direct product of the group of translations of R and the largest connected group of isometries of M". This second factor is SO(n) or the identity component of the Lorentz group 0(1, n-1) according as the curvature is positive or negative. It is easy to see that a discrete subgroup of Z(M) commuting with 0 is generated by a translation of R and by — /e 0(n) if the curvature is positive and by a translation of R if the curvature is negative. In other words, if M is not simply connected, then M = x 5n -1, M =R x i_ 1 (R), M = x (R) or M is a product of R with an (n —1)-dimensional hyperbolic space. We consider now the case (3) of Lemma 2. We shall show that g is a sul;algebra of the Lie algebra o(1, n) of the Lorentz group 0(1, n). Let

pn_

o(1,n)=f -Fp be the Cartan decomposition, j. e., is the Lie algebra of a maximal compact subgroup St of 0(1, n) and p is the orthogonal complement of with respect to the Killing form. The symmetric space associated with this Cartan decomposition is a hyperbolic space. Choose a St-invariant inner product in p, i. e., an invariant Riemannian metric on the associated symmetric space, in such a way that the sectional curvature is —1. Then, if X, Y and Z are three vectors in p, then R(Y, Z) X = [X, [ Y, Z]] = (X, Y) Z —(X, Z) Y,

3. Riemannian Manifolds with Large Group of Isometrics

53

where R is the curvature and ( , ) is the inner product in p. Choose a unit vector X in p and let be the 1-dimensional subspace of p spanned by X. Let p" be the orthogonal complement of nt' in p. Define a subspace nt" of o(1, n) by tn" = (Z+ [X, Z]; Zep"). Let t be the subalgebra of defined by

It is not difficult to see that the subalgebra 1)+ne +nt" of o(1, n) thus defined is isomorphic to the Lie algebra g in (3) of Lemma 2. In verifying this assertion, one should choose X in Lemma 2 in such a way that the constant c is equal to 1 and also make use of the relation [X, [ Y, Z]]=(X, Y) Z— (X, Z) Y above. The correspondence

Z + [X, Z] em"

ep"

defines a linear isomorphism between p =n11 + p" and m =m1 +m". On the one hand, p is identified with the tangent space of the symmetric space 0(1, n)/$t at the origin and has a natural inner product ( , ) corresponding to the invariant Riemannian metric of curvature —1. On the other hand, nt can be identified with the tangent space of the homogeneous space M= E0/6 at the origin and has an inner product ( , which corresponds to the given Riemannian metric of M. Under the isomorphism between p and nt, these two inner products ( , ) and ( , )' may not correspond to each other. But they are none the less closely related to each other. Since ( , is invariant by SO (n —1) (which is the subgroup of SO (n) leaving the 1-dimensional subspace nt' of m invariant), there exist positive constants a and b such that )'

(Z + [X,

(X, X)1 = a(X, X) =b(Z, Z) Z + [x,

for X e m' for Ze p".

In other words, if we define a new inner product ( , )" on m by

(X, X)" = 71 (X, X)' (Y, Y)" =(Y, Y)1

for X E nt' for Ye ni",

then the corresponding new invariant Riemannian metric on M= eqb has constant negative curvature. We claim that M is simply connected. Although this may be proved in the same way as a similar assertion in the case (2), it is an immediate consequence of the general result that a

IL Isometries of Riemannian Manifolds

54

homogeneous Riemannian manifold with negative curvature must be simply connected (see Kobayashi-Nomizu [I, vol. 2; p. 105]). Consider now the case of Lemma 3. Then M is clearly locally symmetric and flat. If M is simply connected, then M =R8 and 0 is a semidirect product of the group of translations and Spin (7) in an obvious manner. (The group of proper motions of R8 is a semi-direct product of the group of translations and SO(8). Since Spin(7) is a subgroup of SO(8), 0 is a subgroup of the group of proper motions of R 8 in a natural manner.) From the fact that Spin (7) acting on R8 is absolutely irreducible, it follows that the identity transformation is the only motion of R8 which commutes with 6. Hence, M must be simply connected. What we have proved may be summarized as follows. Theorem 3.3. Let M be an n-dimensional Riemannian manifold with n >4 such that its group 3 (M) of isometries contains a closed connected subgroup lowing:

of dimension 5 n(n-1)+1. Then M must be one of the fol-

(1) M =R x V, where V is a complete simply connected space of constant curvature and 0=R x 3° ( 7 ); (2) M =51 x V, where V is a complete simply connected space of con(V); stant curvature and 0=51 x (3) M=R x 11_ i (R) and 0 =R x 3° (11_ i (R)); (4) M=S' x g_ 1 (R) and 0=S' x 3°01_, at»; (5) M is a simply connected homogeneous Riemannian manifold 0/5 with a 0-invariant unit vector field X and admits a 0-invariant Riemannian metric of constant negative curvature (which agrees with the originally given metric on the tangent vectors perpendicular to X). If n=8, then the following additional case is possible: (6) M =R 8 and (6=R 8 Spin (7) (semi-direct product), where R 8 denotes the translation group on R 8 and Spin(7) is considered as a subgroup of the rotation group S 0 (8). Remark. A precise description of 15 in (5) as a subgroup of the

Lorentz group 0(1, n) is given in the discussion preceding the theorem. Its Lie algebra is described in (3) of Lemma 2. A local version of Theorem 3.3 is essentially due to Yano [2] although he excluded the case n=8 from consideration. The global version given here is essentially due to Kuiper [3] and Obata As we have shown above (see the paragraph preceding Lemma 1), if a four-dimensional group of isometries acts on a 3-dimensional Riemannian manifold, the action is transitive. E. Cartan ([8; pp. 293-306]) has classified all such groups together with their actions.

55

4. Riemannian Manifolds with Little Isometries

In the 4-dimensional case, difficulties arise from the fact that SO(4) is not simple. The 4-dimensional homogeneous Riemannian manifolds have been studied by Egorov [10] and Ishihara [1]. An extensive work on the dimension of the automorphism group of a Riemannian or affinely connected manifold has been done by Egorov [1, 15]. For a survey of the results on this subject obtained before 1956, see Yano [3]. Mann [1] has shown that if M is an n-dimensional Riemannian manifold, then 3(M) contains no compact subgroups of dimension r for 1(n — k)(n—k+1)+1-k(k+1)0 wherever X *0. Hence, X must vanish at x. Since f attains a local maximum at x, this means that X vanishes in a neighborhood of x. By apply-

ing "Complement to Theorem 1.2" to the local 1-parameter group q. e. d. generated by X, we see that X vanishes everywhere on M. The following result is due to Frankel [3 ]. With a stronger assumption than in Theorem 4.1, we can prove a little more. Theorem 4.4. Let M be a compact Riemannian manifold with nonpositive sectional curvature and with negative definite Ricci tensor. An isometry f of M which is homotopic to the identity transformation must be the identity transformation..

1, be a homotopy such that ho is the identity transformation and h1 =f Let fi be the universal covering space of M Proof Let h„ :).t

with covering projection it Let h, be the unique lift of h, such that It o is the identity transformation of M. (By a lift of h„ we mean h,: Ït( —la such that it 0 hi =h, o it.) We set f= Til . Then f is a lift off and, hence, is an isometry of M. Since each transformation h, normalizes the group of deck-transformations which is a discrete group, the 1-parameter family h, of transformations must commute with the deck-trans:

58

II. Isometrics of Riemannian Manifolds

formations elementwise. Since the deck-transformations are all isometries of JÇI it follows that, for every p ek, the distance d(, f()) between p and f() depends only on it(). Hence, since M is compact, d(, f()) attains an absolute, hence relative maximum at some point po of M. We wish to calculate the Hessian and then the Laplacian of this function d (AM» at po Let c be the geodesic from po to f (p 0 ) ; since k" is simply connected and complete with non-positive sectional curvature, this geodesic exists and is unique (see, for example, Kobayashi-Nomizu [1; vol. 2; p. 102]). For each unit tangent vector X0 at po perpendicular to the geodesic c, we define a Jacobi field X along c extending X0 as follows. Consider the geodesic exp (s X0), IsI 0 if c has positive length. i=1

5. Fixed Points of Isometrics

59

Thus, if c has positive length, then / (Ili , ili)>0 for some Iii , which contradicts the fact that c is the longest of the curves. Hence, c must have zero length, i. e., f (p 0)= p 0 . Since the distance (,f (p)) attains a relative maximum (actually an absolute maximum) at po , it follows that f (p)= il• in a neighborhood of po and hence everywhere on M. q. e. d. We have assumed that in the homotopy ht each k is a diffeomorphism of M onto itself. The full strength of compactness of M was not used in the proof; the theorem still holds if the function F(p)=distance (3, f()), where p = it (P), attains a relative maximum at some point of M. The function F on M plays the same role as the length of an infinitesimal isometry X in the proof of Theorem 4.1 or 4.3. This function has been systematically exploited by Ozols [1 ]. For non-differentiable versions of some of the results in this section, see Busemann [2 ]. The following result of Atiyah-Hirzebruch [1] implies that a compact manifold M with nonzero Â-genus cannot admit a Riemannian metric for which dim 3 (M)> 0, i. e., its degree of symmetry is zero. Theorem 4.5. If a circle group acts differentiably on a compact manifold M, then the Â-genus of M vanishes. 5. Fixed Points of Isometrics The following elementary result shows that the fixed point set of a family of isometries is a nice differential geometric object. Theorem 5.1. Let M be a Riemannian manifold and 45 any set of isometries of M. Let F be the set of points of M which are left fixed by all elements of 45. Then each connected component of F is a closed totally geodesic submanifold of M. Proof Assuming that F is non-empty, let x be a point of F. Let V be the subspace of Tx (M) consisting of vectors which are left fixed by all elements of 15. Let U* be a neighborhood of the origin in Tx (M) such that the exponential mapping expx : U* .—lti is an injective diffeomorphism. Let U =expx ( U*). We may further assume that U is a convex neighborhood. Then it is easy to see that U n F =expx (U* n 1/). This shows that a neighborhood U n F of x in F is a submanifold exp x ( U* n 1/). Hence F consists of submanifolds of M. It is clear that F is closed. If two points of F are sufficiently close so that they can be joined by a unique minimizing geodesic, then every point of this geodesic must be fixed by 15. Hence, each component of F is totally geodesic. q.e.d. Remark. More generally, if M is a manifold with an affine connection and 15 is a set of affine transformations of M, then the set F of points left

60

IL Isometries of Riemannian Manifolds

fixed by 0 is a disjoint union of closed, auto-parallel submanifolds (see Kobayashi-Nomizu [1, vol. 2; p. 61]). The following result shows that the number of connected components in F is limited. Corollary 5.2. In Theorem 5.1, assume that M is complete. Let p and q be points belonging to different components of F. Then q is a cut point of p. If sY• is a connected Lie group of isometries, then q is a conjugate point of p. Proof If q is not a cut point, then by definition there is a unique minimizing geodesic, say c, from p to q. If f is any element of 0, then f (c) is also geodesic from p to q with the same arc-length as c. Hence c = f (c) and every point of c is left fixed by f Since f is an arbitrary point of 0, this shows that c is contained in F and, hence, p and q are in the same connected component of F. Let 0 be a connected Lie group with positive dimension. Assume that q is not conjugate to p. Let c be a geodesic from p to q. Every infinitesimal isometry X defines a Jacobi field along c; the one-parameter group generated by X defines a variation of c in a natural manner. If X belongs to the Lie algebra of 0, then X vanishes at p and q. Since q is not conjugate to p, X must vanish at every point of c. Thus, the 1-parameter subgroup of 0 generated by X leaves c fixed pointwise. Since 0 is connected and is generated by these 1-parameter subgroups, leaves c fixed. This shows that p and q can be joined by a curve c contained in F. q. e. d.

Theorem 5.1 can be strengthened if 05 is a 1-parameter group. Indeed, we have the following result (Kobayashi [5]). Theorem 5.3. Let M be a Riemannian manifold and X an infinitesimal isometry of M. Denote by Zero (X) the set of points of M where X vanishes. Let Zero (X). H v -N. be the decomposition of Zero (X) into its connected components. Then: (1) Each Ni is a closed totally geodesic submanifold of even codimension. (2) Considered as afield of linear endomorphisms of T(M), the covariant derivative V X (= — A x) annihilates the tangent bundle T(N) of each Ni and induces a ( skew-symmetric) automorphism of the normal bundle V (N1 ) of each Ni . Restricted to each N1 vx is parallel, Vv (V X)=0 for every ye T(1Vi). (3) The normal bundle T I (Ni) of Ni can be made into a complex vector bundle. (4) If M is orientable, then each Ni is orientable.

5. Fixed Points of Isometries

61

Proof Applying the proof of Theorem 5.1 to the local 1-parameter group of local isometrics, we see that each Ni is a closed totally geodesic submanifold. Let x e /Vi ..If we choose a suitable orthonormal basis for T(M), then the linear endomorphism X)), of Tx(M) is given by a matrix of the form 0

0 ak k 0 —a Write T(M) = T + Ti', where T (resp. T:) is the subspace spanned by the first n — 2k elements (resp. the last 2k elements) of the basis of T(M). Then Tic is left fixed pointwise by exp (t A) and hence by exp(t X). It is clear that . 7 is the tangent space T;,(/Vi) and Tic ' is the normal space Tx1 (1Vi). This proves (1) and the first statement of (2). According to Proposition 2.2, we have Vy (A x) = R(X, Y) for every vector Yon M. This may be rewritten as for Ye T(M). Vy (VX) = R (X , Y) —

At every point of Zero(X), the right hand side R(X, Y) vanishes and hence Vy (VX) for every vector tangent to M at a point of Zero(X). This proves a little more than what is claimed in (2). Since the eigen-values + i al , ..., +ak of (VX)„ defined above remain constant on each Ni because VX is parallel on I we can decompose the normal bundle T i(J) into subbundles E1 , , Er as follows: ,

T i (lVi)= E1 + - - - + Er

(orthogonal decomposition),

where El , , Er correspond to the eigen-values — b?, , br2 of (VX)x restricted to T 1 (/Vi). Let J be the endomorphism of T i (Ni) defined by —

J I E.=

1 -

bi

(VX). Then J 2 =



I, and J defines a complex vector bundle

structure in T i (lVi). Since T(M) I N = TOO+ T i (lVi), if T(M) is orientable, q. e. d. T(N1) is also orientable. Since the fixed point set F of isometrics or the zero set of infinitesimal isometrics consists of totally geodesic submanifolds, it is perhaps appropriate to mention the following result on totally geodesic submanifolds at this point.

II. Isometrics of Riemannian Manifolds

62

Theorem 5.4. Let N be a totally geodesic submanifold of a Riemannian manifold M. If X is an infinitesimal isometry of M, its restriction to N projected upon N defines an infinitesimal isometry of N. In particular, if M is homogeneous and N is a closed totally geodesic submanifold, then N is also homogeneous.

For the proof, see Kobayashi-Nomizu [1, vol. 2; P. 59], where other properties of totally geodesic submanifolds are also given. This is probably an appropriate place to mention the following topological result. Theorem 5.5. Let M be a compact Riemannian manifold and X be an infinitesimal isometry of M. Let Zero (X). H U -M be the decomposition of the zero set of X into its connected components. Then (1) (- ok dim Hk (M; K)=E(E(— 1)k dim lik (Ni ; K)), t k (2)E dim Hk (M K)> EE dim Hk(lVi; K)) for any coefficient field K. k Proof We shall prove (1) following Kobayashi [5]. The proof of (2) is harder and is omitted (see Floyd [1] and Conner [1]). Let A i be the closure of an s-neighborhood of N . We take e so small that every point of Ai can be joined to the nearest point of N, by a unique geodesic of length e and that Ai n Ai be empty for i j. Then A i is a fibre bundle over Ni whose fibres are closed solid balls of radius s. Set A =-- U A. Let B be the closure of the open set M —A. Then A n B is the boundary of A. We remark that if Uk Vk

1 -4 Vic —

'47k

"

is an exact sequence of vector spaces, then (

_ 1)k di m uk _E( _

dim

_ dim wk = 0

vk

We apply this formula to the exact sequences of homology groups (with coefficient field K) induced by

B

-

11/1



(M,B)

and

AnB



A *(A,AnB) —

and obtain

X (B) — X(M)+X(M, B)=0

and

where x denotes the Euler number. By Excision Axiom, (M, B) and (A, A n B) have the same relative homology. Hence, x (M, B)= x (A, A n B). It follows that

5. Fixed Points of Isometrics

63

Since the 1-parameter group generated by X has no fixed points in B nor in A n B, Lefschetz Theorem implies x(B)= x (A n B) = O. Hence, x(M)= X(A). Since Ai is a fibre bundle over Ni with solid ball as fibre, we have Finally, we obtain

X(M=E X(Ai)=E X(N)We should remark that in Floyd [1] and Conner [1] (2) is stated as follows. If T is a total group acting on a manifold M with fixed point set F, then for any coefficient field K we have Edimilk (M; K)>EdimHk (F; K). To see that their statement means (2), let T be the closure of the 1-parameter subgroup of 3(M) generated by X. Then T is a connected compact abelian group and hence a toral group. Clearly, F =Zero(X). q.e.d. As a generalization of (1) of Theorem 5.5, we mention the following result. Let M be a compact Riemannian manifold and f be an isometry of M. Let F be the fixed point set off If we denote the Lefschetz number off by L(f) and the Euler number of F by x(F), then

L(f)= X (F)-

To prove this statement, we have only to replace Zero (X) by F and the Euler numbers x (M), x (A), ... by the Lefschetz numbers L(f), L( f IA), .... This result has been proved by Huang [1] when f is periodic. We shall see that (1) of Theorem 5.5 is a very special case of Theorem 6.1 in the next section. It becomes also a special case of the classical theorem of Hopf when the zeros of X are isolated since an infinitesimal isometry has index 1 at each of its isolated zeros. For various homological results on periodic transformations and toral group actions, see Borel [3] and the references therein. An infinitesimal version of the following theorem is due to Berger [1]. The generalization given here is due to Weinstein [1]; he proved the result for a conformal transformation. The idea of the proof is similar to that of Frankel [2]. Theorem 5.6. Let M be a compact, orientable Riemannian manifold with positive sectional curvature. Let f be an isometry of M. (1) If n =dimM is even and f is orientation-preserving, then f has a fixed point. (2) If n =dim M is odd and f is orientation-reversing, then f has a fixed point.

II. Isometries of Riemannian Manifolds

64

Proof. Let d(p, f (p)) be the distance between pe M and f (p)E M. It is a non-negative function on M. Let po be a point of M where this function d(p, f (p)) achieves a minimum. We must show that the minimum is zero. Assume that d (Po , f (pa is positive. Let c be a minimizing geodesic from Po to f(p 0). Let N and N' be the normal space to c at po and f (p o), respectively. We shall show that f maps N onto N'. We consider a 1-parameter family of curves cs , —eResf (1Vi).

pgi

Proof The first formula we are going to prove is the following:

( 1)

tdf(tfl-EVX)=L x (f(ti2+VX)),

on M,

where tx is the interior product by X. We shall do all our calculations on the principal bundle P so that X and VX are also lifted to P in a natural manner. We denote by•D the exterior covariant derivation. Then td(f(tS2+VX))=tD(f(ti2+VX))=tD(f(t0+VX,...,t0+VX)) =mtf(DVX,t0+VX,...,t0+VX) =mtf(t x f2,t0+VX,...,t0+VX) =mtf(t x (tS2+VX),t0+VX,...,ti2+VX) =t x f(tfl+VX). In the proof above of (1), we made use of the formula DV X =ix 0; this formula is equivalent to the formula in (1) of Proposition 2.2 but can be derived easily and directly from L x w=0 and t x co =DX, where co is the connection form. We define a 1-form on M-Zero (X) by

( Y) = g (X, Y)/g (X, X)

for Ye T ( M-Zero (X)).

II. Isometries of Riemannian Manifolds

70

Then

(2)

1/J(X)=1,

and

Lt/,=O

i1 ch/J=0.

We set

n=1 (tfl+VX)

1—t

where 1/(1— tchk) means 1 +tdiii + t 2 (d 111) 2 + We prove the following formula:

(3)

AtS2+VX)+tdn—i x ti=0.

This is a consequence of (1) and the following two formulae:

tdri=tdf(tO+VX)

tfr +hi-2+DX) tchk 1—tdtk

1 —t

and

1

ixn=ixi(to+vx) 1---t +1(112 +VX) 1—t di ' d

'

which follows from (2). We are now interested in the coefficient of r in the formula (3). We may write

± where tik is a (2k + 1)-form. (Since dim M =2m, the coefficient of ti‘ vanishes if k is greater than m). Then

txtl=txt/m-Ir -1 ±•••• On the other hand, the coefficient of Hence, we have

(4)

f(Q)+dtim - 1 .0

r

in f(tf2+VX) is given by f(C2).

on M-Zero(X).

Let N denote the 6-neighborhood of N. Using (4) and Stokes formula, we obtain If(Q)=1im S f(Q)= —lim E-■

E-,

M—UNic

M—UNic

.Etim S nm - 1 me E_po •

To complete the proof of the theorem, we have to show that Resf (lVd=lim e.o

6. Infinitesimal Isometrics and Characteristic Numbers

71

Since we are now interested in each individual Ni , we denote Ni by N. The boundary NV, of the e-neighborhood of N is a sphere bundle over N with fibre S2 '- ', where 2r is the codimension of N. Let

aivs—,N we denote by ooNj and 0 (N) the algebras of aNE and N, respectively, then we have a natural CE :

be the projection. If differential forms on algebra homomorphism

a::

0 (N)

0(3Nc).

-

On the other hand, the integration over the fibre in denoted by al . Hence ce* : 4,03 NE) * 0(N)

aisle

-

q■1 will be

-

is a linear mapping which decreases degrees by 2r - 1. Moreover, characterized by the following formula:

(alve),

S u • a: v = I a; u • v for u e

ON ,

N

as.

is

v e 0 (1V) .

If cp is a form defined on M-Zero (X), then we denote by afi, ( (p) the integral over the fibre of the restriction of cp to a1\. We set E.o

provided the limit exists. To calculate the integral lim

sn

E-.0 ai s T.

„,

-1'

we first integrate ri„,_ 1 over the fibre in aNs —)N and then integrate the result over the base N. Hence, we are interested in o-* (ri 1 ). But this is the coefficient of tm-' in a. ?I. We are now interested in calculating =0-* (f (t Q +V X)

tfr

1



\

t dill I .

Since f (t C2 +DX) is smoothly defined on the entire space M including N, it follows that til \ a* n=f(to +vx)IN . a* (1 — t dill I ' (5) The problem now is to evaluate a* (Oil - t d 114 Since Ill =tp+tikAdt/J-Ft 2 tfrA(dt/4 2 +•••, 1-tdtk

the problem is further reduced to that of calculating a* (4/ A (d Or 1.

II. Isometrics of Riemannian Manifolds

72

Let be the 1-form corresponding to the infinitesimal isometry X under the duality defined by the metric tensor. Then

=

2 IIII '

and di I i .

g0 2 g —dC112) A g 11 4

so that

tii A (d tlir - 1 =

(6)

gii 2 4 • We fix a point o of N and let x1 , ..., x 2 p, y', ..., y 2 r , m=p+r, be a normal coordinate system around o such that N is locally defined by yi ....=y 2 r=0. Such a normal coordinate system exists since N is totally geodesic. We consider the Taylor expansions of and cg at o. Then

(7) (8)

=E A u y i dyi+•••, g=EA u dy i Adyi-Ek iab A ik yk yi dxa Adx b +--,

where the dots indicate terms with total degree in y and dy greater than 2 and (At,) is a skew-symmetric non-degenerate matrix. These two formulae may be proved as follows. If we denote y i. , ... , y2 r by x2 P+1 , ..., x2 p +2 r and write =E A dx A, then

(A)=O 4; B± B; A" &4;B;C+ (41)0

E RAB

=0

(2) of Proposition 2.2, D=

0 (1) of Proposition 2.2,

unless A, B._., 2 p+ 1

(&4;B)A,B=2 p+1,...,2 p+2r (2 p

is non-degenerate at o

E RO a b dxa A dxb) =0 a, b=1 o Or

(Theorem 5.3), (Theorem 5.3),

unless C, D -2p C,

D

2p +1

(Nis totally geodesic).

From these formulae, both (7) and (8) can be easily obtained. It is clear that in calculating a* (lk A (4)4-1 ) we can replace and cg by their Taylor expansions and ignore the terms of degree sufficiently high in y and dy. It is indeed not difficult to see that the terms indicated by dots in (7) and (8) can be ignored. (For the detail on this point, see Bott [2; pp, 321-323 ]. )

6. Infinitesimal Isometries and Characteristic Numbers

73

We set =E Aii y i d yj, 13=EAu dy 1 dyi, y=ER; ab A ik yk yi dxa Adx b

so that and d c are approximated by a and # — y, respectively. From (6), we obtain

a A (fi

(9)

0 04 2 11

(

I

Since the dimension of the fibre in aivE -3 N is 2r — 1, the integration over the fibre annihilates any term whose degree in d y is not exactly 2r —1. This reduces (9) to o.* (1// A (d o )" -1)=0 (10)

0-* (0 A(4)41-1)=

for q al + • • • + a„ then e l (M) is negative (see Hirzebruch [1; p. 159]) and,

by Theorem 2.1, the group 5(M) is finite. In particular, if M is a nonsingular hypersurface of degree greater than n +2 in Pn+1 (C), then 5(M) is finite. It is of some interest to note that the hypersurface M in Pn+i (e) n+1

E (z =o in terms of a homogeneous coordinate system io z °, .. . , f + 1 is not hyperbolic for any degree d, provided n 2. In fact, such a manifold contains a rational curve: defined by

)"

(u, /0 eP1 (C) —) (u, y, w u, w y, 0, ... , 0)EP„ + ,(C), where w denotes a d-th root of —1. On the other hand, I know of no example of a compact hyperbolic manifold whose first Chem class is not negative. If M is of the form DIE', where D is a bounded domain in C" and I' is a properly discontinuous group of holomorphic transformations acting freely on D, then M is hyperbolic and also ci (M) is negative. For the proof of the first assertion, see Kobayashi [9,10]. Let 2 E goc„ de d? be the Bergman metric of D. Since it is invariant by I', it may be considered also as a metric on M =DIT. Similarly, the 2-form —L E g Œ, .de A d? 2ni may be considered as a 2-form on M. From the definition of the Bergman metric, it is clear that this 2-form represents the first Chem class ci (M). Since (go) is positive definite, c 1 (M) is negative, thus proving the second assertion. The holomorphic transformation group 5(M) of M =DIT is therefore fmite either by Theorem 2.1 or by Theorem 2.2. The finiteness of 5(M) for M =DIE has been proved by Bochner [2], Hawley [1], Sampson [1 ]. In connection with Theorem 2.1 and one of the examples above, we mention the following result. Theorem 2.3. Let M be a non-singular hypersurface of degree d in + 1 (C). If n 2 and d 3, then the group 5(M) of holomorphic transformations of M is finite, except in the case when n=2, d= 4. See Matsumura and Monsky [1 ], where a completely algebraic proof is given. Lemma 14.2 in Kodaira-Spencer [1] shows also that dim $3(M)=0 if n 2 and d. 3. Matsumura and Monsky show that 5(M) can be an infinite discrete group when n =2 and d= 4. The reader will find also a completely algebraic proof of Theorem 2.1 in Matsumura [1 ]. We say that an algebraic manifold M of dimension n is of general type if 1 sup lim —„ dim H° (M,Km)>O, m. + co tn-

III. Automorphisms of Complex Manifolds

88

where K denotes the canonical line bundle of M. The following theorem generalizes Theorem 2.1. Theorem 2.4. If M is a projective algebraic manifold of general type, then its group 6(M) of holomorphic transformations is finite. For a completely algebraic proof of this theorem, see Matsumura [1 ]. A transcendental proof can be also given along the same line as the proof of Theorem 2.1. We again map 6 (M) onto a group of linear transformations of the vector space H=1/° (M, ICP) leaving a certain star-like bounded domain D invariant. The only nontrivial part of the proof is to show that this representation is faithful if p is large. But this follows from the result of Kodaira to the effect that we can obtain a projective imbedding of M using a certain subspace of H = H° (M, Kt') for p large. (For the detail, we refer the reader to the Addendum in KobayashiOchiai [2 ]. ) For a compact Riemann surface we have the following very precise result of Hurwitz [1 ]. Theorem 2.5. Let M be a compact Riemann surface of genus p 2. Then the order of the group of holomorphic transformations of M is at most 84(p 1). -

We shall only indicate an outline of the proof. Let V be a compact Riemann surface of genus p' and f: M —0/ an n-fold covering projection with branch points. Let aeM be a branch point. With respect to a local coordinate system z with origin at a and a local coordinate system w with origin at f (a), the mapping f is given locally by w=2" around a. Then m— 1 is called the degree of ramification of f at a. Let al , ... , ak be the branch points of f with degrees of ramification ml , ..., m k . Then the Riemann-Hurwitz relation states k

i.1

where x (M) and x (V) denote the Euler numbers of M and V. This formula can be easily verified by taking a triangulation of V such that f (a 1 ), . . . are vertices and the induced triangulation of M and then by counting the numbers of vertices, edges and faces. Let eo be a finite group of holomorphic transformations of M; we know already that the group of holomorphic transformations of M is finite if the genus p of M is greater than 1. Let %. denote the isotropy subgroup of 45 at aEM. Let f: M —) MA5 be the natural projection. If the order m of 05a is greater than 1 and if z is a local coordinate system around a eM, then we introduce a local coordinate system w around

2. Compact Complex Manifolds with Finite Automorphism Groups

89

f(a)e114/0 by z= wm. In this way, MAD becomes a compact Riemann surface which we shall denote by V. Then M is a branched covering of V with projection f, to which we apply the Riemann-Hurwitz relation. The degree of ramification of f at a is equal to m-1. If we denote by n the order of 15, then the s-orbit through a consists of n/m points. The sum of the degrees of ramification of f at these points on the s-orbit 1 -— . Hence, the Riemannof a is therefore equal to rTi (m —1) = n (1 m Hurwitz relation is of the form X(M)+n

1 (1-- ) =n • X(V)•

>

mi Since mi is the order of a subgroup of 6, mi divides n. If we denote by p' the genus of V, then the formula above may be rewritten as follows: 2p — 2 If

=2p' 2+

E (1 1=1'

1 ). mi

2, then (p —1)/n1. If p' =1, then (2p— 2)/n> (1 —

and hence n 4(p —1). Finally, consider the case p' =O. Then

(2p— 2)/n= — 2+

E

1 (1-- )=k- 2—

It follows that k3. If k5, then (2p — 2)/n

2

2

k

E 2

1 mi = 2

1 mi

and n

For k =4, we have the following possibilities:

>2 =2 =2 =2

m2

m3

7774

>2 >2 =2 =2

>2 >2 >2 =2

>2 >2 >2 >2

(2p-2)/n >2

= 3. >1.. =2 > =T1 =>1.

ri 3(p— I) ir 4(p-1) 71. 6(p-1) n 12(p— 1)

For k= 3, we have the following possibilities: .m2

>3 =3 .3 =2 =2 =2

>3 >3 =3 >4 =4 =3

7713

(2p-2)/n

>3 >3 >3 >4 >4 >3

>+ > =w1 =12 1

8(p-1) n12(p-1) n24(p-1) n -20(p 1) 72-40(p 1) n 84(p 1) —





90

III. Automorphisms of Complex Manifolds

Let M be a compact complex manifold and K be its canonical line bundle. Let k be a positive integer such that Kk is very ample over some nonempty open set U of M in the following sense. At each point x of M, (M; K I`) consisting of holomorphic let H(x) be the subspace of H = sections of K i‘ vanishing at x. Assume that, for each x e U, H(x) is a hyperplane of H and that the mapping x e U —) H(x) gives an imbedding of U into the projective space P(H*) (see the proof of Theorem 2.1). Then the natural representation p of 5(M) on H is faithful. In fact, if a holomorphic transformation of M leaves every holomorphic section of Ki` fixed, then it leaves every point of U fixed and, being holomorphic, it leaves every point of M fixed. In particular, if M is a compact Riemann surface of genus p 2, then K is very ample over some nonempty open subset U, and it follows that a holomorphic transformation of M leaving every holomorphic 1-form fixed is the identity transformation. Thus we have the following result of Hurwitz DI Theorem 2.6. A holomorphic transformation of a compact Riemann surface of genus p 2 is the identity transformation if it induces the identity transformation on the first homology group Hl (M ; R). For higher dimensional analogs of Theorem 2.6, see Theorem 4.4 of Chapter II and Borel-Narasimhan [1]. For more results on automorphisms of compact Riemann surfaces of genus 2, see Macbeath [1], Lehner-Newman [1 ], Accola [1, 2], Lewittes [1 ]. For an analog of Theorem 2.5 for algebraic surfaces, see Andreotti [1]. Somewhat related with the results of this section is the following theorem of Gottschling [I]. Let H. be the Siegel upper-half space of degree m, i.e., the space of complex symmetric matrices of degree m with positive definite imaginary part and 11 be the Siegel modular group of degree m. Then, for m 3, the group of holomorphic transformations of H,./F,, consists of the identity element only. 3. Holomorphic Vector Fields and Holomorphic 1-Forms If Z is a holomorphic vector field and co is a holomorphic 1-form on a complex manifold M, then w(Z) is a holomorphic function on M. If M is compact, this function must be constant. This simple fact yields some useful results. Proposition 3.1. Let M be a compact complex manifold and b =1)(M) the Lie algebra of holomorphic vector fields on M. Denote by W i'° the space of closed holomorphic 1-forms on M. Define (Zeb; co(Z)=0 for all co EV.°)

91

3. Holomorphic Vector Fields and Holomorphic 1-Forms

Then (1) ih is an ideal of lj and contains [f),1)], (2) dim bib, bi , where 1;1 is the first Betti number of M. Proof We recall the following general formula relating a 1-form w and vector fields Z and W: 2 dco (Z, W) = Z(o)(W))— W(co (Z))

([Z, W]).

(1) follows immediately from this formula. Let be the space of closed anti-holomorphic 1-forms. It is a simple matter to verify that V. ° (resp. W°. 1) is the space of closed (1, 0)-forms (resp. (0, 1)-forms). Let V be the space of closed (complex) 1-forms. Then

Let 21 be the space of cobounding 1-forms, i.e., 1-forms of the type d f. Then

Let

g. 01. ° = {coeV. ° ; co(Z)=0-for all Zeb}, 0°. 1 = fro ee. 1 ; 45 (2-) = 0 for all Z eb) =P,° .

Then the pairing (a),

z) e w1,0 x 9 —> co(Z)eC

induces a dual pairing between V. °/01. ° and filth. We shall show 0 ± %90,1) Œ g1,0 ± g0,1 .

Let d f = a+ #effl1 n w1,0 +w0,1•) where a, fieV. °. Then df+df=a+fi+d+fi. Let Zelj. Then (df + df)(Z)=(a+ fi)(Z). Since a and fi are holomorphic, the right hand side is constant. On the other hand, the left hand side vanishes at the maximum point of the real valued function f + f Hence, (Œ+ f3)(Z)=0. Similarly, from we obtain

(df —df)(Z)= (a — /3)(Z), (a— f3)(Z)=0.

Hence, a(Z)=13(Z)= 0. This proves our assertion that df is in 01 . °

gO, I.

92

III. Automorphisms of Complex Manifolds

We may now conclude 2. dim fillyi =2 . dim V. °/gl a °= dim (WL ° + W°. 1)/(gl. ° + 0°' 1) dim(" o w o,

n

dim V/R1 = .

q. e. d.

o

Remark. A holomorphic vector field Z with non-empty zero set belongs to the ideal bp

The ideal 1), of 1)(M) was introduced by Lichnerowicz [3] to study b(M) of a compact Kdhler manifold M with non-positive, non-negative or zero first Chern class. See also Matsushima [5]. 4. Holomorphic Vector Fields on 'Uhler Manifolds Let M be a Kdhler manifold of dimension n. Let Z be a complex vector field with components COE, C' in terms of a local coordinate system z", i.e., 2, We can write

VZ=V 1 Z+V"Z,

where V1 Z and V"Z are defined by the property that

ViarZ =0 and V41, Z =0

for all vectors W of type (1, 0).

In terms of a local coordinate system,

a

viz=Evft ccedzPo Ta-z—a +Evpcadzfico v"z=

vp

a a? Ta-z,r +E

di'

a a

a?



Similarly, for any tensor field K, we can write

VK=V1 K+V"K. Given a complex vector field Z of type (1, 0), we denote the (0, 1)-form corresponding to Z by C. In terms of their components, we have Z = E cce

a 4—* c=E Cp d?

with Co = E gap Ca.

Proposition 4.1. A complex vector field Z of type (1,0) on a Ktihler manifold M is holomorphic if and only if V"Z =0, or equivalently, V" C=0 (where is the (0, 1)-form corresponding to Z).

4. Holomorphic Vector Fields on Kahler Manifolds

93

Proof. Clearly, Z is holomorphic if and only if, for each point p of M, there exists a local coordinate system z', zn around p such that (0Cyail) p = 0. On the other hand, since M is a Kahler manifold, for each around p such point p there exists a local coordinate system z', that (V p .(alag) (i.e., such that the Christoffel symbols vanish at p). Hence, Z is holomorphic if and only if Vp COE= 0. q. e. d. )

Theorem 4.2. Let M be a Kahler manifold, Z a complex vector field of type (1,0) and the corresponding (0, 1)-form. If Z is holomorphic, then PC=z1"C=ER ŒA CŒdZfi. Conversely, if M is compact and

g(z1"C—E R ŒA CŒde,C)dv=0, then Z is holomorphic. Proof. In Appendix 3, the following general formula for a (0, 1)-form

is proved: A 11 = E(—va vecco nfl+Rap ca de). Since VŒ=

gjA Vy

Proposition 4.1 implies V' CA =0 if X is holomorphic. Hence, we obtain the first statement of the theorem. To prove the converse, we use the following integral formula (see Theorem 4 of Appendix 2) expressed in terms of a local coordinate system:



" c)# cfl +

Rap. ca cfl + »cce • VTG} d v 0 .

The first two terms of the integrand cancel each other by our assumption. Hence, Cc • VIZ-, d v = O.

This implies » COE = 0. By Proposition 4.1, Z is holomorphic.

q. e. d.

The first half of the theorem is due to Bochner [1] and its converse to Yano [4].

Theorem 4.3. Let M be a compact Kahler manifold and Z= X —iJX a complex vector field of type (1, 0) with real part X. Then X is an infinitesimal isometry if and only if Z is holomorphic and div X =0. Proof. We use the characterization for an infinitesimal isometry obtained in Theorem 2.3 of Chapter II. By Theorem 4. 2 Z is holomorphic if and only if X satisfies (1) in Theorem 2.3 of Chapter II. q. e.d.

III. Automorphisms of Complex Manifolds

94

Theorem 4.3 is due to Yano [4 ]. Theorem 4.4. Let M be a compact Kahler manifold and Z a holomorphic vector field (of type (1, 0)) with the corresponding (0, 1)-form C. Then (1) C=H C + d" f, where H1 is the harmonic part of and f is a ( complex valued) function. Such a function f is unique if it is normalized by the property fdv=0.

(2) (=d" f if and only if œ(Z)= 0 for every holomorphic 1-form a, i.e., if and only if Z e fh, where b l is the ideal introduced in § 3. (3) The real part X of Z is an infinitesimal isometry if and only if the real part off is a constant. (This means that if f is normalized as in (1), then f is purely imaginary.)

Proof By Proposition 4.1, d" (=O. Now, (1) follows from the HodgeKodaira decomposition theorem: C=HC-Fd"6"q)+6"d"(p, where q) is a form with the same bidegree as C. If C = H C + d" f = H C + d" g, then d"(f — g)= 0, that is, f— g is holomorphic and hence a constant. To prove (2), we observe first that if a is a holomorphic 1-form, then Œ(Z) is a holomorphic function on M and hence is a constant. It suffices therefore to prove that the integral j. a(Z) dv vanishes for all holomorphic 1-forms a if and only if H C = O. Assume H C= 0. Then a (Z)= g (a, C)= g(Œ, d"f), where g denotes the inner product on the cotangent spaces defined by the metric. Since a is holomorphic, it is harmonic. Since a harmonic form is perpendicular to d'f=d"f in the Hodge-Kodaira decomposition, the integral g(a, d" f)dv vanishes. Assume, conversely, that œ(Z)= O and let a= H C; since H? is a harmonic (0, 1)-form, its complex conjugate is a harmonic (1, 0)-form and hence is holomorphic. Then

0= 11((Z)dv= J (g(HC,H C)+g(II C,d"f))dv= J g(II C,H ()dv. Hence, H=0. X is an infinitesimal isometry if and only if div X =0 (Theorem 4.3). On the other hand, div X =0 if and only if 6(C +Z) =0. But

= (ô " d" + d" (5") f + (ô ' d' +d' 6') f =d" f + f =Of +Itlf=1,61(f+f).

5. Compact Einstein-KOhler Manifolds

95

Since d(f + J7)= 0 if and only if f + f is a constant, we obtain the assertion q. e. d. in (3). (I) and (2) of Theorem 4.4 are due to Lichnerowicz [6]. Corollary 4.5. In Theorem 4.4, if the zero set of Z is nonempty, then 1=d" f. Proof. If a is a holomorphic 1-form, then œ(Z) is constant. If the zero

set of Z is nonempty, then a (Z)= O. Our assertion follows from (2) of Theorem 4.4. q. e. d. Corollary 4.6. In Theorem 4.4, assume that the real part X of Z is an infinitesimal isometry. Let C be the (0, 1)-form corresponding to Z and be the real 1-form corresponding to X. Then the following statements are mutually equivalent: (1) The zero set of Z ( =the zero set of X) is nonempty; (2) C = d" f, where f is a function with purely imaginary values; (3) c =Jdu, where u is a real valued function.

5. Compact Einstein-KHhler Manifolds In this section we shall prove the following result of Matsushima [I]. Theorem 5.1. Let M be a compact Einstein-Kahler manifold with nonzero Ricci tensor. Then the Lie algebra i(M) of infinitesimal isometries is a real form of the Lie algebra 14(M) of holomorphic vector fields, i.e., 1)(M)=i(M)+{:i • t(M). In the statement above, i(M) is imbedded in t(M) by identifying an infinitesimal isometry X with the corresponding holomorphic vector field Z=X—iJX (see Theorem 4.3). Proof. By our assumption, the Ricci tensor RŒ and the metric tensor &co satisfy the relation:

Ro=c • go ,

where c is a nonzero constant.

Let Z be a holomorphic vector field (of type (1, 0)) and C the corresponding (0, 1)-form. By Theorem 4.2, A" C = c C .

Substituting C = H + d" f (see Theorem 4.4, (1)) into this, we obtain A" d" f=c(HC+ d" f),

III. Automorphisms of Complex Manifolds

96

which shows that El( =O. We may assume that f is normalized as in Theorem 4.4 and we write f=u+i v, where u and v are real valued functions. We shall show that

.4" d"u=cd"u

LI" d"v=cd"v.

and

Since ( =d"f and d" C = c C, we have

0= d"d"f cd"f=d"4"f d" cf=d"(A" f—cf), —



which shows that d"f —cf is a holomorphic function and hence is a constant. But

d" f —cf =(A"u—cu)+/(61" v—cv)=(4z1u—cu)+4.1z1v—cv). Hence, both the real part (z1"u— c u) and the imaginary part (X v— c v) of d"f —cf must be also constant. It follows that d"(z1"u — c u)= 0 and d"(z1" v— c v)=0, showing our assertion. By Theorem 4. 2 this means that the vector fields U and V of type (1,0) corresponding to the (0, 1)-forms du" and i d" v are holomorphic. By (3) of Theorem 4.4, —iU and V correspond to infmitesimal isometrics since —iu and iv are purely imaginary. Since C = d" f = d" u + i d" v, we have

Z= i —i U)+ V. (

This shows that b ( w). i(m)+1/=-T. • i(M).

q.e. d.

In the course of the proof, we established that d"f — cf is a constant. Integrating d"f —cf over M and observing that the integral of z1"f (=id f) vanishes, we see that this constant is zero if f is normalized. Hence, ef=cf, or zlf 2 c f..= Denote by "2 , the set of all complex valued functions f which are eigen functions of the Laplacian LI with eigen value 2c, i. e.,

Ac= {f ;

df= 2cf}

.

Then the correspondence

f—C=d"f—)Z gives a complex linear isomorphism between g$72 , and b(M). The subspace of F2, consisting of purely imaginary functions corresponds to i(M). Since b (M)= 0 if c < 0 by Theorem 2.1, Theorem 5.1 is of interest only when c> 0.

6. Compact Kdhler Manifolds with Constant Scalar Curvature

97

Theorem 5.2. Let M be a compact Kahler manifold with vanishing Ricci tensor. Then the Lie algebra 1)(M) of holomorphic vector fields coincides with the Lie algebra i(M) of infinitesimal isometries. It consists of parallel vector fields and is abelian. Proof. Let Z be a holomorphic vector field and c the corresponding (0, 1)-form. By Theorem 4.2,

is harmonic. In the decomposition =1-1C+d"f in (1) of that is, Theorem 4.4, the function f is zero (if it is normalized). By (3) of Theorem 4.4, the real part of Z is an infinitesimal isometry. This establishes 13(M)= i(M). By Corollary 4.2 of Chapter II, every infinitesimal isometry of M is a parallel vector field. Clearly, every parallel vector field is an infinitesimal isometry. Finally, the general formula [X, Y] —(Vx Y — Vy X) = 0 implies that the Lie algebra of parallel vector fields is abelian.

q. e. d.

For a generalization of Matsushima's theorem to compact almost Einstein-Kahler manifolds, see Sawaki It is interesting to find out how large the class of compact EinsteinKahler manifolds is. It is not known if there exists a non-homogeneous compact Einstein manifold. In this connection, see Berger [2] for a survey on Einstein manifolds and Aubin [1] for a construction of certain Einstein-Kahler metrics. 6. Compact Kkhler Manifolds with Constant Scalar Curvature

In this section we shall prove a theorem of Lichnerowicz [2, 3] which generalizes Theorem 5.1 of Matsushima. Theorem 6.1. Let M be a compact Kahler manifold with constant scalar curvature. Let b(M) denote the Lie algebra of holomorphic vector fields, i(M) the Lie algebra of infinitesimal isometries (considered as a subalgebra of 1)(M)), c the subalgebra of b(M) consisting of parallel holomorphic vector fields and b the ideal of b(M) consisting of those vector fields Z such that a(Z)=0 for all holomorphic 1-forms Œ. Then (1) 13(M) =b + e ( Lie algebra direct sum); (M) b), i.e., i(M)n b is a real form of b, (2) b = (1(M) r b) + (3) i(M) = (i(M) n b) + c. Proof. We denote by Q the linear endomorphism on the space of (0, 1)-forms defined by QC=E RŒff COE dg for

III. Automorphisms of Complex Manifolds

98

Let Z be an arbitrary holomorphic vector field on M and C the corresponding (0, 1)-form. Following Theorem 4.4, we write

C= 9 + d"f,

where 9= H1 =the harmonic part of C.

As in § 5 we write f = u+ iv, where u and y are real valued functions. We set = d" u and ti = id" v so that We shall show that 9 corresponds to a vector field belonging to c and that c and q correspond to vector fields belonging to {:i(i(M)r) b) and i(M) r) b, respectively. We shall make use of the following formula which follows easily from the second Bianchi identity: Lemma. If we denote by R the scalar curvature, then

EveRap=vp-R

and

EvoR=va R.

Since 9 is a harmonic form of degree (0, 1), its conjugate is holomorphic and hence V1, 941 = O. This fact, together with Lemma, implies d" (Q 9)= —Evo(R„, (pa)=—E 9Œ Da R=0. Since Z is holomorphic, Theorem 4.2 implies

d"C—QC=0. Since d" 9= 0, this implies f)— Q(d" f)= Q q. Hence, S (d"(d" f)— Q(d" f), d" f) dv= 5 (Q9, d" f) dv

= 1 (5" (Q 9), f) dv= O. By Theorem 4.2, this means that the vector field corresponding to the (0, 1)-form d" f is holomorphic. Hence, the vector field corresponding to 9 (=C— d" f) is also holomorphic. By Proposition 4.1, Vii 941 = O. Since we already know that Vo 941 =0, we can conclude that 9 is parallel. Since d" f = + n corresponds to a holomorphic vector field, Theorem 4.2 implies d" — Q = — (d" ri — Qt». Hence 6" (d" — Q

)

= - (5" (A" q — Q q) -

99

6. Compact KKhler Manifolds with Constant Scalar Curvature

We shall show that both sides of this equality vanish by demonstrating

that the left hand side is real and the right hand side is purely imaginary. We have Q )= (5"d" 6" — (5" (Q = d" (5" d" u+ E vo(RŒA • Vau) =.4" • tru+ER„ A •VP- Vau.

This shows that 6" (A" c — Q 0 is real. Similarly,

ROE , • vo va v),

(5" (A" — Qq). i(z1" A" v +

which shows that 6" (A" — Qq) is purely imaginary. Hence, Now we have (.4"

dv= j(z1"—N,d"u)dv =

By Theorem 4.2, corresponds to a holomorphic vector field. Hence, corresponds also to a holomorphic vector field. Denote by Zo , Zi , Z2 the holomorphic vector fields corresponding respectively. Then to the (0, 1)-forms cp, Z = Zo + + Z2 .

We have shown already that Zo is in c. By (2) and (3) of Theorem 4.4, Z1 is in {:1(i(M) r b) and Z2 is in i(M) n b. The facts that

b n c= 0 and (i(M) n

(i(M) b)) = 0

n

are also immediate consequences of Theorem 4.4. To prove that [b, =05 let

z=

Vœf

a aza

eb

and W=

Then

Ez,141= (vaf. Vcc (PP — 49P VI3V œf) =—

(9P vavp f) 2 - aza

a

(pa

a azŒ

e c.

= — E koft VP Vœf)

E \Ng" vp f)

a aza

a

III. Automorphisms of Complex Manifolds

100

since W is parallel. But

E9-73v,I=E0,1v.f. is a harmonic (0, 1)-form, Eço,dzP

Since E 9A dg is a harmonic (1, 0)form and hence is a holomorphic 1-form. We set oc= dzP. On the

a

other hand, the vector field Z' =E vPi is also in b. Since ot(Z)= azP by Theorem 4.4, we obtain

v" f.o. This completes the proof of the fact that [Z,W]= 0. Finally, (3) follows from (1) and the fact that every element of c, being parallel, is in i(M). q. e.d.

7. Conformal Changes of the Laplacian In order to study compact Kahler manifolds with non-negative or nonpositive first Chan class, it is convenient to introduce the Laplacian with respect to a hermitian metric conformal to the given Kahler metric. The results in this section are due to Lichnerowicz [6]. We follow both Lichnerowicz [6] and Matsushima [5]. Let M be a Kdhler manifold and let e be a real, positive function on M. We introduce operators 45; and A operating on complex differential forms by a" (ea .0

(5:', = e

for every differential form ço,

and A :,1 = o

+ d" o 5:.

By a direct calculation using local coordinates (see Appendix 3), we obtain (51;9=6"9-1(dio -)9,

where t(di a) denotes the interior product of 9 with the complex vector field of type (0, 1) corresponding to the (1, 0)-form d'a. We obtain easily 4.= — (d" 0 t(d' a)-F t(d 1 a) 0 d").

If M is compact, we define a new inner product ( , )a by (9, Oa = S (9, tTi) • e • dv

Then (d" (P)

cT&P) 13';

7. Conformal Changes of the

Laplacian

101

In fact, (d" 9, ky = (d" 9, e tT)= (9, 6" (e til)) = (9, e'' 6" (e 111))0 = (9,

A differential form 9 is said to be 4-harmonic if di:, 9=0. Theorem 7.1. Let M be a compact Keihler manifold and denote by the space of 4-harmonic (p, q)-forms and by OP the sheaf of germs of holomorphic p-forms over M. Then

dim H:.q< oo;

(1) (2)

In particular, dim He" is independent of a.

The result (1) is due to Kodaira and (2) is the result of Dolbeault. For the proof and further references, see Hirzebruch [1; Chapter IV]. We define a tensor field with components Co by Co = Rap VilVa a, —

and denote by Q a the linear endomorphism on the space of (0, 1)-forms defined by Qa C = E C (HP for C= E Ca dr . Then the following theorem generalizes Theorem 4.2. Theorem 7.2. Let M be a Kiihler manifold, Z a complex vector field of type (1, 0) and C the corresponding (0, 1)-form. If Z is holomorphic, then 4C—Q a (C)=0. Conversely, if M is compact and

(A 1(11 C — 02„(0,

= 0'

then Z is holomorphic. Proof We prove the following lemma which is a local formula.

Lemma. A

— Q a (C)= d"C — Q(C)—V s C, where Q= Q 0 and S is the vector field of type (0, 1) corresponding to the (1, 0)-form d'a.

a

Proof ofLemma. Since d' a =EV„a dza and S=EVa Ta--F , we obtain

d" 0 t(d 1 a) = d"(E V i a. CO= E (v„ vet c„ + vet a • VA CO dg, t (d' a) 0 d" = E Va cr (Va Cp— Vp CO dg.

III. Automorphisms of Complex Manifolds

102



Hence,

461::C=A"C—(d" 0 t(d' a) - F t(d' cr) 0 d") C = A" C — E eir a - VŒ C0 + V0V a • ca) (1513 = A" C—Vs C—EVA Va a -CŒ difl .

Now the lemma follows from the following fact:

Suppose Z is holomorphic. Since S is of type (0, 1), Proposition 4.1 implies Vs ( =0. Then by Lemma,

The first half of Theorem 7.2 follows from Theorem 4.2. To prove the second half, we make use of the following integral formula (see Appendix 2):

(A " C — Q(O, 5) = (v" C, V" co), where C and co are arbitrary (0, *forms. We set w= e c. Then the left hand side is equal to (A" C — Q (0, 0, while the right hand side is equal to

1 (E VA C' V(e. Ca)) dT/= 1 (E VACŒ • VA a • Ca + E vp-cce • Vif CŒ ) e • d1/ , CL. =(VsC)k+ (V" CV" Hence,

(A" C — Q(0 — v s C, 0 , = (v" c,v" 0, .

By Lemma, this formula may be written as (A icri C — Qff (0) 0, = (V" Now, if the left hand side vanishes, then V"C = 0. By Proposition 4.1, Z q. e. d. is holomorphic. Let dv denote the volume element of a Uhler manifold M and define a positive 2n-form Q= e dv. Let bc, be the Lie algebra of holomorphic vector fields leaving the form 0 invariant, i.e.,

bc,= (z eb(M); Lz fl = 0) . k

Proposition 7.3. For a compact lathier manifold M, the subalgebra of the Lie algebra b(M) of holomorphic vector fields may be defined by

= (Z et)(M); 44 C=0) , where C denotes the (0, 1)-form corresponding to Z.

8. Compact Kahler Manifolds with Nonpositive First Chern Class

103

Proof We have

Lz 0=Lz (ea)dv+e' • L z dv=L z a - 0—ea - 6"C • dv = (I (d'a) C — 6" C) fl = — 6:1,C - O.

This shows that LO= O if and only if = O. By Proposition 4.1, d" ( = 0 whenever Z is holomorphic. Hence, for an element Z of 1)(M), 6'; ( = 0 is equivalent to 6'; C= d" C = O. This in turn is equivalent to d; C=0. q.e. d. Theorem 7.4. Let M be a compact Kahler manifold. Then the sub1)(M) possesses the following properties: algebra t)

(1) b , is abelian; (2) if Zell, and co(Z)=0 for all holomorphic 1-forms co, then Z=0 (that is, ber n th = 0, where th is the ideal of 1)(M) introduced in § 3); (3) dim 1)0. .1- b 1 , where b 1 is the first Betti number of M; (4) if Z is a nonzero element of lx„ then the zero set of Z is empty.

Proof We recall that 1)1 is the ideal of 1)(M) consisting of vector fields Z such that w(Z)= O for all holomorphic 1-forms co. (Since M is a Kdhler manifold, every holomorphic form is closed.) Since 1)1 contains the derived algebra of t) (M) (see Proposition 3.1) and since f) , is a subalgebra, we have

[IL , IL] OE IL n 1)1 .

Let Zel),,,n bi• By (2) of Theorem 4.4, we have C=d"f. Hence,

(C, Oa = fri"f, 4, = (i; 6'; Oa = ° by Proposition 7.3. Therefore, Z= 0, thus proving (2) and hence (1). Now (3) follows from Proposition 3.1 and the inclusion IL = ILAI), n 1)1) c= b(m)/bi .

Suppose Zek and Zero(Z)#Ø. As we remarked at the end of § 3, Z must be in 1)1 . By (2), Z = O. This proves (4). q.e. d. 8. Compact Khler Manifolds with Nonpositive First Chern Class

Let M be a compact complex manifold and c1 (M) its first Chern class. We say that c 1 (M) is nonpositive and write ci (M) 0 if it can be represented by a closed (1, 1)-form y = 2i, i

E Co de A d9

III. Automorphisms of Complex Manifolds

104

such that (Co) is negative semi-de fi nite hermitian. We have already considered manifolds with negative first Chern class (see §2). Assume that M is a compact KMler manifold with Ricci tensor R Then c1 (M) is represented by p=---

27r

E RŒ

dZŒA d.V.

By Theorem 1 of Appendix 4, a (1, 1)-form p is cohomologous to y if and only if there exists a real valued function a such that Cao = Rap — VOE a. We prove two theorems of Lichnerowicz [6] (see also Matsushima

[5]). Theorem 8.1. Let M be a compact lahler manifold with c1 (M).0 and t) (M) be the Lie algebra of holomorphic vector fields on M. Then (1) b(M) is abelian and dimb(M)-1-13 1 . (2) If Z is a nonzero element of t) (M), it never vanishes on M. i (3) If a closed (1, 1)-form y =n E Cap dza A dZI3 represents ci (M) and 2 (Ce) is negative semi-definite everywhere and negative definite somewhere, then 1)(M)= 0. Proof Let Z be a complex vector field of type (1,0) with the corresponding (0, 1)-form C. By Theorem 7.2, Z is holomorphic if and only if

(z1:1,C — Q7(c),

=0.

This is equivalent to

c„„ cec v= 0. Since (Co) is negative semi-definite, this equation is equivalent to

d"C=0,

(5 .C=0,

CŒp COE = O.

In other words, Z is holomorphic if and only if C is i1-harmonic and satisfies E COEfi ca = 0. By Theorem 7.1, dim f)(M)_._ dim H°. 1 (M ; C). If (Ca p) is negative definite at a point p and Z is holomorphic, then C must vanish in a neighborhood of p and hence everywhere on M. Let 1)1 be the ideal of f)(M) consisting of holomorphic vector fields Z such that a(Z)=0 for all holomorphic 1-forms Œ. In general, 1)(M)/f)1 is abelian (Proposition 3.1). We shall show that f 1 =0. Let Z efh. By

8. Compact Kghler Manifolds with Nonpositive First Chan Class

105

Theorem 4.4, C=d"f, where f is a function. On the other hand, C is harmonic. Hence, C=0. Since '4, =0, 13 (M) itself is abelian. Suppose the zero set of Z is nonempty. If Z is holomorphic, for every holomorphic 1-form a the function Œ(Z) is holomorphic and hence constant. Since a(Z) vanishes at some point, it must vanish identically. This means that Z is an element of 13 1. q. e. d. Hence, Z=0. Corollary 8.2. Let M be a compact Kahler manifold with c 1 (M) 0. If r is the maximal rank of (Cap), then dim1)(M)n—r, where n is the complex dimension of M. Remark Theorem 8.1 implies that if cl (M)0 almost everywhere. Let 9 be a holomorphic n-form on M. Since every holomorphic form on M is necessarily closed, we obtain Lz 9=do t z 9+t z od9=0.

Since 4=(d9)=0 and tz =0, we obtain Lz rp=dot z rp-Ft z odrp=0.

Hence, L z ((p A (7).o.

III. Automorphisms of Complex Manifolds

116

Making use of this formula, we obtain

-f L z (9 i.\)=L z (f.

Z(f)•4 yN=L z (f7. 9 A

Since f. 9

A

A4

0 is a form of degree (n, n), it is closed and hence Lz (f9 A ço)=d o t z (f. (

0)=

A-

by Stokes' theorem. Hence,

Z(

(in2 9 A 0=0.

(The factor in2 makes the (n, n)-form in2 Q y.‘i) real and non-negative.) Since Z(f) ,--- I1Z112 is positive almost everywhere, we conclude that 9 q.e.d. vanishes identically on M. Since a holomorphic n-form is a holomorphic section of the canonical line bundle K, the following theorem generalizes that of Howard. Theorem 11.2. Let M be a compact Kiihler manifold with a nontrivial holomorphic vector field Z such that Zero (Z) is nonempty. Let K be the canonical line bundle of M and p be a positive integer. Then the line bundle KP admits no nonzero holomorphic sections.

= d"f be the (0, 1)-form corresponding to Z as in the proof of Theorem 11.1 so that Z(/)= II Z11 2. Let 9 be a holomorphic section of KP. As in the proof of Theorem 2.1, in terms of a local coordinate system 9 may be symbolically expressed as follows: Proof Let

= h(di

A

•••

A d?)",

where h is a holomorphic function defined in the coordinate neighborhood. Define a real, non-negative (n, n)-form 19 I 2/P by

I91 2/P = r2

dz i A •-•

A

de A cre A

•••

A

dr

In the proof of Theorem 2.1 we established that the largest connected group of holomorphic transformations of M leaves every holomorphic section of IQ invariant Hence 1912/P is invariant by the 1-parameter group generated by Z, that is, L z (191 2/P). O. The remainder of the proof is essentially the same as the proof of Theorem 11.1. Since

Z019113/2 = Lz(1149r 2)-1 Lz(k'r 2)=Lz(f 191P12) =do tz(f 1(pI P12). Integrating the both ends of the equalities above and making use of Stokes' theorem, we obtain

Z(/)191 142 =O.

11. Zeros of Holomorphic Vector Fields

117

Since Z(t). Zil 2 is positive almost everywhere, we conclude that I (pIP I2 and hence ço vanish identically on M. q.e.d. The following theorem of Howard sharpens Theorem 11.1 for Hodge manifolds. Theorem 11.3. Let M be a Hodge manifold with a nontrivial holomorphic vector field Z such that Zero (Z) is nonempty. Then M admits no nonzero holomorphic p-form for p> dim Zero (Z). The set Zero(Z) is a subvariety (possibly with singularities) of M, and by dim Zero (Z) we mean the maximum dimension of its components. Proof Let n = dim M. By Theorem 9.4, there is an imbedding of M into a complex projective space PN (C) such that the holomorphic vector field Z can be extended to a holomorphic vector field, denoted also by Z, of PN (C). Since the automorphism group of PN (C) is finitely covered by the group SL(N +1; C), Z may be considered as an element of the Lie algebra of SL (N +1; C). With respect to a suitable basis of CN + 1, ziv be the coordiZ is represented by a triangular matrix, and let z°, nate system in CN + 1 with respect to such a basis. Let Lk be the k-dimensional linear subspace of PN (C) defined by

Then Z is tangent to L k at each point of Lk for k =0, 1, , N. Let a) denote the Kahler (1, 1)-form of PN (C) corresponding to an invariant Kahler metric. Multiplying w by a suitable positive constant, we may assume that w represents the characteristic class of the line bundle over PN (C) defined by a hyperplane. Thus w represents the second cohomology class dual to the 2(n-1)-dimensional homology class represented by a hyperplane. Lemma 1. Let M and Z be as above. If p is a nonzero holomorphic p-form on M, then there exist r-dimensional irreducible subvarieties If, of M for r =n, n— 1, , p with the following properties: (1) Z is tangent to n at every regular point of V,. P] denotes the (2) If j,: 17,—)M denotes the imbedding and [cp A cohomology class in H2 ' P (M ; C) represented by the (r,r —p)-form rp A P, then 4 [q• A_P] is a nonzero element of 112 r - P(K; C). The proof of Lemma 1 is by induction on r starting with r=n. For r= n, it suffices to take K= M. It is well known that the multiplication by (On- P defines an isomorphism of HP(M; C) onto 1/2 ""(M; C) (see for example Weil [1; p.75]). Hence Up A CO" - II] is nonzero. Assuming Lemma 1 for r, let k be the smallest integer such that Lk contains 1/,..

III. Automorphisms of Complex Manifolds

118

Viewing V, as an algebraic variety in the k-dimensional projective space Lk , we consider the hyperplane section V, n Lk_ i . Since co is the characteristic class of the line bundle defined by a hyperplane, we obtain [9 A al - P] I(Vr 4-1)=[(1) A al- 11 I Vr•

The right hand side is nonzero by assumption. We write V, n Lk_i = m i Di , where Di's are irreducible divisors of V, and mi's are integers. Since [9 A for - 1- P] I (V, n Lk _ i)*0, it follows that [(I) A of -1- I Di * 0 for some Di . We denote such a divisor Di by I'. clearly J' the properties (1) and (2) for r — 1. This completes the proof of Lemma 1. Let V be the p-dimensional irreducible subvariety vp obtained in Lemma 1 and V' be the set of regular points of V. Denote the inclusion map V'—>M by j. Then as in the proof of Theorem 11.1, we have

E

Z(f) • j*(9A0)=j* L z (f7pArp)=j*dot z (f(pnço)+j* tz o d(Ao Ac -p-) j* do t z (fp ço). Now we make use of the following Stokes theorem (see P. Lelong [1; § 6]): Lemma 2. Let V be a p-dimensional st4bvariety of a compact complex manifold M and V' be the set of regular points of V. Let 0 be a (2p —1)form defined in a neighborhood of V. Then d0=0.

Hence,

V.

Z(f) • j* ((p A F.p)= j* d 0 t z (ho

rp)= O.

As in the proof of Theorem 11.1, we conclude that 9=0 since Z cannot vanish identically on V' by our assumption dim Zero (Z)< p =dim V'. Corollary 11.4. Let M be a Hodge manifold with a holomorphic vector field Z whose zero set Zero(Z) is nonempty and discrete. Then (1) M admits no nonzero holomorphic p-forms for p >0; (2) the arithmetic genus of M is equal to 1; (3) the fundamental group ni (M) has no proper subgroup of finite index and Hl (M; Z) --- 0. Proof (1) is immediate from Theorem 11.3. (2) follows also imme-

diately since the arithmetic genus of M is equal to

E( —1)P dim HP. ° (M; C)= dim H°. ° (M; C)= 1 . To prove (3), assume that ni (M) has a subgroup of finite index k and let it71 be the corresponding k-sheeted covering space of M. Lift Z to a

12. Holomorphic Vector Fields and Characteristic Numbers

119

holomorphic vector field 2 of ia and apply (2) to A. Then the arithmetic genus of /a is also 1. On the other hand, the Riemann-Roch-Hirzebruch theorem implies that the arithmetic genus of /a is k times that of M. Hence, k =1. Since Hi (M; Z)=ni (M)1[7r1 (M), ni (M)] and Hi (M; R). 0, it follows that [ni (M), 7E1 (M is a subgroup of ni (M) of finite index and hence, Hi (M; Z)= O. q. e. d. (3) of the corollary above is due to Lichnerowicz [8]. He has shown many other results on zeros of holomorphic vector fields on compact Kahler manifolds with non-negative first Chem class. )]

12. Holomorphic Vector Fields and Characteristic Numbers Let M be an n-dimensional compact complex manifold and Z be a holomorphic vector field on M. Let Zero (Z)=U 1Vi be the decomposition of the zero set of Z into its connected components Ni . We assume that Z is non-degenerate along iv, in the following sense. Since we consider one iv, at a time, we denote iv, by N. First, we assume that N is a complex submanifold of codimension r so that dim N=n—r. Let Ti .°(M) and T1' ° (N) be the holomorphic vector bundles of complex vectors of type (1, 0) over M and N, respectively. We set E . mi.° I (MIN, E` = o (N) . Then E is a holomorphic vector bundle of rank n over N and E' is a holomorphic vector subbundle of E of rank n — r. The holomorphic vector field Z induces an endomorphism of the bundle E; in terms of a local coordinate system zl, , zn of M it is given by the matrix

(acc y a zi)

if Z= E

a

We denote this holomorphic endomorphism 0 zOE of E by A. (Since Z vanishes on N, the matrix (acc zP) defines A independently of the coordinate system.) The kernel of A contains E'. We assume that E' is exactly the kernel of A. Then denoting the image AEcE by E", we obtain a decomposition —.

E= E' ()E". Thus, E is holomorphically isomorphic to the direct sum of two holomorphic vector subbundles E' and E". We denote the restriction of A to E" by A. Then A is an automorphism of E". Let h' and h" be hermitian fibre metrics in E' and E", respectively. We extend the hermitian fibre metric h' +h" of E to a hermitian metric h of M. We follow now closely § 6 of Chapter II. Let P be the bundle of unitary frames over M; it is a principal bundle over M with group U(n).

III. Automorphisms of Complex Manifolds

120

Let PM be the bundle of adapted frames over N; it is a principal bundle over N with group U (n — r) .x U (r). (By an adapted frame we mean a unitary frame whose first n—r basis elements are in E' and whose last r elements are in E".) Considering the hermitian connection of M, let Q be the curvature form on P. Then its restriction to 4, is of the form IQ,

0\

ko

Qv !'

the curvature form for h' where, with an obvious identification, Q and Qv is the curvature form for h". Let V denote the covariant differentiation of the hermitian connection defined by h. As in § 4, we write VZ=V/Z+V"Z, where V' Z and V"Z are defined by the property that Viv Z=0

and V,1;, Z=0

for all vectors Wof type (1, 0).

Then the endomorphism A of E coincides with the restriction of V' Z to N. As in §6 of Chapter II, we may consider V1 Z as a tensorial 0-form of type ad( U(n)). Restricted to 4,, v'Z is of the form /0 0\ \0 A Let f be an ad(U(n))-invariant symmetric form of degree n on the Lie algebra u(n). The simplest example is given by det (=determinant). As in §6 of Chapter II, we define the residue Resf (N) by Resf (N) • tn - ' =

f(ti2+SPZ) det(tS2,+A)'

where the bars over f and det indicate that the forms are pull down to N. Now the theorem of Bott [1, 2] may be stated as follows:

Theorem 12.1. Let M be a compact complex manifold of dimension n. Let Z be a holomorphic vector field of M with zero set Zero(Z)= H N -

where the Ni's are the connected components of Zero(Z). Let f be an (ad U(n))-invariant symmetric form of degree n on u(n). Then the characteristic number I f(Q) of M defined by f is given by SAM= E Resf (Ni), i

provided that Z is non-degenerate along Zero (Z) in the sense defined above.

12. Holomorphic Vector Fields and Characteristic Numbers

121

The proof is almost identical to that of Theorem 6.1 of Chapter II and hence is omitted. We remark that A= VT ZIN is holomorphic and hence the coefficient of tk in f(tf2+V"Z)I N is a polynomial in the curvature form with constant coefficients since a holomorphic function on a compact manifold must be constant. (In § 6 of Chapter II, a similar fact was derived from the property that VX is parallel on N.) In Bott [1], the theorem is proved when Zero(Z) consists of isolated points. The general case is proved in Bott [2]. A generalization to meromorphic vector fields has been obtained by Baum-Bott [1]. For a different proof, see Atiyah-Singer [1]. When Zero(Z) consists of isolated points, the theorem follows also from a Lefschetz formula of AtiyahBott [1]. See also Illusie [1]. Corollary 12.2. If a compact complex manifold admits a holomorphic vector field with empty zero set, then its Chern numbers vanish.

IV. Affine, Conformal and Projective Transformations

1. The Group of Affine Transformations of an Affinely Connected Manifold Let M be a manifold with an affine connection and L(M) be the bundle of linear frames over M. Let 0 and co denote the canonical form and the connection form on L(M), respectively. We recall (§ 1 of Chapter II) that a transformation f of M is said to be affine if the induced automorphism f of L(M) leaves co invariant. We quote the following result established earlier (see Theorem 1.3 of Chapter II). Theorem 1.1. Let M be an n-dimensional manifold with an affine connection. Then the group 91 (M) of affine transformations of M is a Lie transformation group of dimension ..n(n+ 1). As in § 3 of Chapter II, it is natural to ask when the maximum dimension n(n + 1) is attained. We prove Theorem 1.2. Let M and 91(M) be as in Theorem 1.1. Then dim 91 (M) = n (n +1) if and only if M is an ordinary affine space with the natural flat affine connection.

Proof Assume dim 9.1[(M)= n(n+ 1). From Theorem 1.3 of Chapter II,

it is clear that the identity component 9I°(M) acts simply transitively on each connected component of L(M). This implies that every standard horizontal vector field 5C (1. e., vector field iC such that (Da) =0 and 000=a constant element in R") on L(M) is complete; the proof is similar to that of Theorem 2.5 of Chapter II. This means that the connection is complete (see Proposition 6.5 of Chapter III in KobayashiNomizu [1]). Let 91„ be the isotropy subgroup of 91(M) at a point xeM. Since dim 9I„ = n 2 , 9I„ contains GL + (n; R), where

(n; R) = {aeGL(n; R); det a>0}. (Identifying 9Ix with the linear isotropy group, we consider it as a subgroup of GL(n; R) by choosing a basis in the tangent space Tx (M).) In

1. The Group of Affine Transformations

123

particular, it contains homothetic transformations t In with t > O. If K is a tensor of contravariant degree p and covariant degree q at the point x, K. Since the curvature tensor the transformation t in sends K into field R is of contravariant degree 1 and covariant degree 3, the transformation tin sends Rx into t - 2 Rx . On the other hand, 91„ must leave Rx invariant so that Rx = r 2 Rx for all t> O. Hence, Rx = O. This shows that the curvature tensor field vanishes identically. Similarly, the torsion tensor field vanishes identically. Hence, the connection is flat. Let A4 be the universal covering space of M. Since AI is a simply connected manifold with a complete flat affine connection, it is an ordinary affine space with the natural flat affine connection (see Theorem 7.8 of Chapter VI in Kobayashi-Nomizu [1]). From the fact that no element of 91(k) other than the identity commutes with 91° (k) elementwise, we can conclude that M itself is simply connected as in the proof of Theorem 3.1 of Chapter II. Finally, the converse is evident. q.e.d. The local version of Theorem 1.2 is classical; see, for example, Eisenhart [1]. The following theorem is due to Egorov [1] (see also Yano [3]). Theorem 1.3. Let M be an n-dimensional manifold with an affine connection and 91(M) be the group of affine transformations. if dim 91(M) > n 2, then the connection has no torsion. Proof Assuming that the torsion tensor does not vanish at a point

x,

we shall show that the dimension of the isotropy subgroup 21„ is at most n2 — n. Since 9.1„ leaves the torsion tensor at x invariant, it suffices to prove the following algebraic lemma. Lemma. Let V be an n-dimensional vector space and T be a nonzero element of V () A 2 V*, where V* is the dual space of V. Let G be the group of linear transformations of V leaving T invariant. Then

dim G< n2 n. Proof of Lemma. We consider Tas a skew-symmetric bilinear mapping V x V—) V Let X: V—) V be a linear transformation and 91 = et x be the 1-parameter group of linear transformations generated by X. Then X is

in the Lie algebra g of G if and only if T(9, v, 9, w)= 9,(T(v, w))

for v, we V and all teR.

Differentiating this equation with respect to t at t= 0, we see that X is in g if and only if T(X y, w)+ T (v , X w)= W (v , w))

for y, we V.

124

IV. Affine, Conformal and Projective Transformations

Take a basis in 17 and let Tiik and Xi be the components of T and X with respect to the chosen basis. Then the equation above is equivalent to the following system of linear equations:

E (TIc xi+ Tbai — xi Tbjc)=o

for a,b, c=1,

n.

This can be rewritten as follows:

0

(a,b,c=1,...,n),

where aj

i= Tac

Ta b i 61 — Tic (51.

This is a system of linear equations with n 2 unknowns Xj and coefficients i . If we can find n linearly independent equations in this system, then we will know that the dimension of the space of solutions of this system does not exceed n 2 n. Let T:c be one of the non-vanishing components of T Since Tik is skew-symmetric in the lower indices, Tba,*0 implies b * c. By reordering the basis if necessary, we may assume that either a = 1, b = 2, c= 3, or a=b=-1, c=2, i.e., Th 0 or T112 +0. We shall first consider the case where T21 3 + 0. We claim that the n linear equations given by

E

Xj= 0 k=1, ...,n

are linearly independent. In fact, consider the n x n matrix (An defined by 4= Ak213 . Then TA 61+ ni 6; —

— 713 (51:.

Since Th + 0, the matrix (4) is non-singular. Next, assume T112 +0. We claim that the n linear equations given by

EA1LX i ==0

EAui xii =o k=2,..., n, are linearly independent. In fact, consider the n x n matrix (Ai) defined by = A-12 2 = T 2 22 + T122 6i2

2 6i= T 2 12 (5i — 2I pi

J

•31 + — Ti k 61 for j= 1, , n and k= 2, ... , n.

Alc= A112 = Vic (5.-i

Then this matrix is of the following form: —

* 0 Tj2

2. Affine Transformations of Riemannian Manifolds

125

where in_ 1 denotes the identity matrix of order n — 1. This matrix is q. e.d. clearly non-singular. We state Another result of Egorov [3] which can be proved by a similar method (see also Yano [3]).

Theorem 1.4. Let M and 91(M) be as in Theorem 1.3. If dim 91(M) > n2, then the connection has neither torsion nor curvature, j. e., it is flat, provided n.. 4. Remark. In connection with Theorem 1.3, Egorov [5] (see also Yano [3]) has shown that if dim 91(M)= n2 and n 4, then there exists a 1-form T on M such that the torsion tensor T is given by T(X, Y)= t (Y) X — T (X) Y for all vector fields X, Y. For example, define an affine connection in M =Rn in terms of the natural coordinate system x', ..., xn and the Christoffel symbols as follows: EA= —61, r=o for j#1, so that its torsion T can be constructed from the 1-form t = d x l in the manner described above. Since the group 91(M) contains the translations in Rn, it is transitive on M =Rn and its isotropy subgroup 910 at the origin is the group of linear transformations of Rn leaving the point x' =1, x 2 = • • • ---- xn = 0 invariant. Hence, dim 91(M)= n 2. Clearly, the curvature of this connection vanishes identically. If we symmetrize the affine connection above by setting

rtkk =r;,ii = --pi,

r=o for j, k # 1,

then we obtain a torsionfree affine connection with nontrivial curvature eln M = W. The group 91(M) of affine transformations for this connection is the same group of dimension n2. Wang and Yano [1] have determined the n-dimensional affinely connected manifolds such that dim 91(M) > n 2 — n +5. For more details on related results, see Yano [3].

2. Affine Transformations of Riemannian Manifolds In this section we shall compare the group 3(M) of isometries and the group 91(M) of affine transformations of a Riemannian manifold M. The following result is due to Hanno [1]. See also KobayashiNomizu [1, vol. 1; p. 240].

Theorem 2.1. Let M = Mo x Mi x • • • x Mk be the de Rham decomposition of a complete, simply connected Riemannian manifold M, where M 0 is

126

IV. Affine, Conformal and Projective Transformations

Euclidean and M1 , ..., Mk are all irreducible. Then

91° (M ) = 9-01 4 o) x 91° (A4 1) x ... x 3° (M)= 3 ° (M 0) X 3Cb (M1 ) X — X

Since the group of affine transformations and that of isometrics of a Eulcidean space are well understood, the study of 21°(M) and 3°(M) is essentially reduced to the case where M is irreducible. For a proof of the following result, the reader is again referred to Kobayashi-Nomizu [1, vol. 1; p. 242]. Theorem 2.2. If M is a complete, irreducible Riemannian manifold, then 9.1(M)= 3(M) except when M is a 1-dimensional Euclidean space. Let M be a complete Riemannian manifold and AI be its universal covering space. Let ft-4 = Mo x Mi x • • • X Mk be the de Rham decomposition of A1 . By Theorem 2.1, the Lie algebra a(S1) of 91(k) is isomorphic to a (Mo)+ a (MI ) + • • • + a (Mk). Let X be an infinitesimal affine transformation of M and it be its natural lift to M". Let (X0 , X1 , ... , Xk) be the element of a (M0) + a (M1 ) + • • • + a (Mk) corresponding to )-. C E a (M). By Theorem 2.2, X1 , ..., Xk are all infinitesimal isometrics. If Al has no Euclidean factor M0 , then X0 is zero so that it is an infinitesimal isometry. If 'it. is an infinitesimal isometry, so is X. The assumption that Mo be trivial can be expressed in terms of the restricted linear holonomy group. We can therefore state the result as follows (see also Lichnerowicz [1; p. 83], Yano-Nagano [3]). Corollary 2.3. If M is a complete Riemannian manifold such that its restricted linear holonomy group W °(x) leaves no nonzero vector at x fixed, then 2e(M)=3 °(M). The assumption in Corollary 2.3 is technically a little stronger than the condition that M admits no nonzero parallel vector field. The latter condition amounts to assuming that the linear holonomy group tri(x) leaves no nonzero vector at x fixed. With the notation above, if the length of an infinitesimal affine transformation X of M is bounded, the same is true for X0 , X1 , ... , X. But an infinitesimal affine transformation X0 of the Euclidean space Mo is of bounded length if and only if X0 is an infinitesimal translation which is a very special kind of infinitesimal isometry (see Kobayashi-Nomizu [1, vol. 1; p. 244]). Hence (Hano [1]), Corollary 2.4. If X is an infinitesimal affine transformation of a complete Riemannian manifold with bounded length, then it is an infinitesimal isometry.

3. Cartan Connecticins

127

In particular, we rediscover Corollary 2.4 of Chapter 11 due to Yano. Corollary 23. If M is a compact Riemannian manifold, then 91°(M)-3°(M). 3. Cartan Connections

In this section we shall treat conformal and projective connections in a unified manner. Let M be a manifold of dimension n, .2 a Lie group, 20 a closed subgroup of 2 with dim 2/20 =n and P a principal bundle over M with group 20 . We give a few examples: Example 3.1. Let .2 be a Lie group and 20 a closed subgroup of Q. Set M = 2/20 and P== Q. Example 3.2. Let 2 be the affine group 9I(n) acting on an n-dimensional affine space and 20 = GL(n; R) an isotropy subgroup of 2 so that 2/20 is the affine space. Let M be a manifold of dimension n and P the bundle of linear frames over M. Example 3.3. Let .2 be the group of Euclidean motions of an n-dimensional Euclidean space and 20 =0(n) an isotropy subgroup of 2 so that 2/20 is the Euclidean space. Let M be a Riemannian manifold of dimension n and P the bundle of orthonormal frames over M. Example 3.4. Let 2 be the projective general linear group PGL(n; R) acting on an n-dimensional real projective space and 20 an isotropy subgroup of 2 so that 2/20 is the projective space. Let M be an n-dimensional manifold. We shall later construct a principal bundle P over M with group 20, called a projective structure. Example 33 . Let 2 be the Möbius group 0(n +1, 1) acting on an n-dimensional sphere S" and 20 an isotropy subgroup of .2 so that 2/20 is the sphere S. (This will be explained in detail later.) Let M be an n-dimensional manifold with a conformal structure and P be the first prolongation of the conformal structure. (This will be also explained in detail later.) Since 20 acts on P on the right, every element A of the Lie algebra 10 of 20 defines a vertical vector field on P, called the fundamental vector field corresponding to A (Kobayashi-Nomizu [1, vol. 1; p. 51]). This vector field will be denoted by A*. For each element a of 20 , the right translation by a acting on P will be denoted by Ra . A Cartan connection in the bundle P is a 1-form w on P with values in the Lie algebra I of 2 satisfying the following conditions:

128

IV. Affine, Conformal and Projective Transformations

(a) co(A*)= A for every A E 10 ; (b) (R.)* = ad (a -1 ) co for every element a e 20 , where ad (a") is the adjoint action of a' on I; (c) w(X) * 0 for every nonzero vector X of P. Condition (c) means that a) defines a linear isomorphism of the tangent space (P) onto the Lie algebra I for every ueP since dim P. dim Q. In other words, co defines an absolute parallelism on P. A Cartan connection in P is not a connection in P in the usual sense, for a) is not I0-valued. It can be, however, considered as a connection in a larger bundle /32 obtained by enlarging the structure group of P to 2, i.e.,

P2 =Px 20 .2. Then P is a subbundle of P2 and a Cartan connection a) in P can be uniquely extended to a usual connection form on P2, also denoted by a). (Take, for instance, Example 3.2. Then an affine connection of M is a Cartan connection in P. On the other hand, a linear connection of M is an ordinary connection in P. For more details on this point, see Kobayashi-Nomizu [1; Chapter III, §§ 2-3].) If we set E=P2/.20 , then E is the bundle with fibre 2/2 0 associated with P. We can identify M with the image of the natural mapping P P2/20 . In other words, we have a natural cross section M E. In Examples 3.2 and 3.3, E is the tangent bundle of M and the natural cross section is the zero section. In Example 3.1, E is the product bundle M x M and the natural cross section M E. M x M is the diagonal map. Geometrically speaking, condition (c) means that the fibre of E over each point xeM is tangent to M at x, see Ehresmann [1], Kobayashi [4]. But this geometric interpretation of (c) will not be used here. We define the curvature form 0 of the Cartan connection a) by the following structure equation: —

dco.



[co, co] +O.

It is an I-valued 2-form on P. For instance, if a) is an affine connection, then Q is a 2-form with values in the Lie algebra a(n) of the group 91(n) of affine motions. Decompose a (n) into the vector space direct sum of the translation part Rn and the linear transformation part gl(n; R). Then the Rn-component of Q is the usual torsion form and the gl (n; R)-component of Q is the usual curvature form of the corresponding linear connection, see Kobayashi-Nomizu [1; § 3 of Chapter III]. Given a Cartan connection co in P, we call a transformation q) of P an automorphism of (P, co) if it is a bundle automorphism, j. e., commutes with the right translations Ra , (ae 20), and if it preserves the form a).

129

3. Cartan Connections

From Theorem 3.2 of Chapter I we obtain the following result (Kobayashi [4]): Theorem 3.1. Let co be a Cartan connection in P. Then the group 91(P, co) of automorphisms of (P, co) is a Lie group with dim 9I(P, a)). dim P. For a fixed point u0 of P, the mapping Qe 91 (P, co)—> cp(u 0)eP imbeds 91(P, w) as a closed submanifold of P. From now on we shall assume that the Lie algebra I of .2 is graded as follows: (a vector space direct sum), 1 ==g-i+go+gi+•••+9k with and lo = go + + • • • + gk • Egi, g] c9 1 Let w=a)--i+wo+col+•••+cok be the corresponding decomposition of the Cartan connection w. Since w defines an absolute parallelism on P, the algebra of differential forms on P is generated by w (i. e., the components of w with respect to a basis for I) and functions on P. But the curvature Q of w does not involve co o , W 1 , , wk . In fact, this is a direct consequence of the following three facts: (i) The g_ 1 -component w_ 1 , restricted to each fibre of P, vanishes identically. (ii) The I0-component w 0 + w 1 + • • • + Wk, restricted to each fibre of P, is the Maurer-Cartan form and hence defines an absolute parallelism. (iii) The curvature Q, restricted to each fibre of P, vanishes identically. Condition (a) for Cartan connections implies both (i) and (ii). To prove (iii), it suffices to observe that the structure equation of the connection restricted to a fibre gives the structure equation of Maurer-Cartan for the group 20 and then apply (i). If we choose a basis el , , en for g_ 1 and write w_ i =wi el +••• +wn en , then we can express the curvature Q of the Cartan connection w as follows: Q=ElKii coi nari, where each Ku is an I-valued function on P. Theorem 3.2. Let .2 be a Lie group and 2 0 a closed subgroup of .2 such that the Lie algebra I o of .20 contains an element E with the property that the Lie algebra I of .2 is graded as follows:

I= 9 -1 + go + + • • • + 9k

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IV. Affine, Conformal and Projective Transformations

where

gr= {X el; [E, X] = r X}

for r= —1, 0, 1, , k

Let P be a principal 20-bundle over M and co a Cartan connection in P (with values in 1). If, the automorphism group 91(P, co) has the maximum dimension, i. e., dim 91(P, co)= dim P, then the curvature Q of co vanishes identically. Proof: Set a, = exp(tE)e 2 0 . Fix a point 140 e P and let ço, be an element of 91(P, co) such that Rat u0 =q),(u 0). Since the orbit of 9.1(P, co) through 140 is closed and has the same dimension as P, such an element (p, exists.

Let Q, be the g,-component of S2 and write

fl„ =E4 'Cry w t Since ço, leaves w and S2r invariant, it leaves K ru invariant. We shall compare this action of ço, with that of R. t . Since ad (E) coincides with the multiplication by r on g„ ad(a,) is the multiplication by en on g,.. Hence, Ra*,(S r) = ad (a7 1 ) r = " Q„ Rt(co_ 1 )=ad(a,-1 )co_ 1 =e1 (0_ 1 .

From these two equalities, we obtain e - ri K rii

coi=e- rt Or = Rt(Or)

=E Rt(K ri) ncoi A R:coi =E4 R(K ru) e2 f cd A a).

Hence, R t (K ru)=6,-(r+ 2)1 K ru .

On the other hand,

e (K O = K r

(

ij •

Comparing these two equalities at 140 , we obtain u (u0)

= K, u(q), u 0)= K r ii (Rat u0) = e ( r+ 2" K ru (u0) .

Hence, Krii (u0)= O. Since uo is an arbitrary point of P, this proves our q.e.d. assertion. If we apply Theorem 3.2 to Examples 3.2 and 3.3, we obtain local versions of Theorem 3.1 of Chapter II and Theorem 1.2. (Note that l= g_ 1 +g, in Examples 3.2 and 3.3 and the.g_ 1 -component of Q is the torsion form while the go-component of S2 is the curvature form in the usual sense.) For this section, see also Ogiue [2].

4. Projective and Conformal Connections

131

4. Projective and Conformal Connections In the preceding section, we studied Cartan connections in general. In this section we shall describe both projective and conformal connections in a unified manner. We begin with a simple algebraic proposition (see Kobayashi-Nagano [1]). Proposition 4.1. Let 1= g_ 1 + go + g1 + • • • +gk be a semisimple graded Lie algebra (with g * 0 and go #0). Then (1) I=9-1+g0+ 91, i.e., g2 =O. (2) The linear end omorphism a of1 defined by

_1

a(X_ i + X0 + Xl. )= X_ i +

XIL for Xi E gi

is an involutive automorphism of 1.

(3) With respect to the Killing-Cartan form B of!, (i) g_ 1 + g1 is perpendicular to g0 ; (ii) Blg_ i =0 and Blgi =0; (iii) g1 is the dual vector space ofg_ 1 under the dual pairing (X_ 1 , B(X_ i , X1 ). (4) The two representations ad1 (g0)1 g_ and adi (go) I gi of g0 are dual to each other with respect to the Killing-Cartan form B. (5) There is a unique element Eego such that gi ={Xel; [E, X]=iX)

i= —1,0,1.

Proof (1) Let Xegi and Z eg2 . Since (ad X)(ad Z) maps gi into gi+i , 2 and i+ 2 1 so that j*j+ i+ 2, the trace of (ad X) (ad Z) is zero.

Hence, B(X , Z)= 0. Since B is non-degenerate, it follows that Z=0. (2) The proof is straightforward. (3) Since B is invariant by the automorphism a, we have

B(X_ 1 + X1 , X0) =B (a (X_ 1 + X1 ), a (X0)) = —B(X_ 1 + X1 , X0), which proves (i). To prove (ii), let X_ 1 , Y_ 1 eg_ i . Then (ad X_ 1 ) (ad Y_ 1 ) maps gi into g._ 2 and hence its trace is zero. This shows that

Y_ 1 )=0. Similarly, B I gi =0. To prove (iii), let X_ 1 eg_ i and assume B(X_ , g1 )-0. By (i) and (ii), this implies B(X_ 1 ,I)=0. Hence, X_ 1 =0. Similarly, if e gi and B(Xit , g_ 1 )=0, then X 1 =0. (4) Let X0 ego . Since B is invariant by ad X0 , we have

B (ad (X0 ) X_ 1 , X 1 )= — B(X_ 1 , ad (X0) X1 )

for Xi egi , i= +1.

IV. Affine, Conformal and Projective Transformations

132

(5) Let E be the linear endomorphism of I defined by for Xi egt ,

a(Xi)=iXi

i= —1,0, 1.

Then s is a derivation of the Lie algebra. Since every derivation of a semi-simple Lie algebra is inner, there is a unique element Eel such that ad(E)= i. To prove that E is in go , we observe that ad(ot(E)) coincides q.e. d. with ad (E). Hence, Π(E)=: E. By the definition of a, E ego .

The graded Lie algebras of Proposition 4.1 have been classified by Kobayashi-Nagano [1]. We are interested here only in the following two example. For a general theory, see Ochiai [3]. Example 4.1. Let 2/20 be the real projective space of dimension n, where 2= PGL(n; R)= SL(n +1; R)/center, { A0 ESL(n + 1 ; R) } /center, Qo = ( a) where A eGL(n; R) and

is a row n-vector,

21 = { (

y I 0 1 ) ; c: row n-vector } .

The graded Lie algebra I= g_ ii + go +g1 with this .2/20 is given by

1=5I(n+1; R),

9-1= {

Co o)

v },

go= I(

A 0 ); trace A + a=0}, 0 a

where y is a column n-vector, The element Eeg o is given by

E=

lain 0\

k o bi '

g1=

{(0 00» ,

is a row n-vector, A egl(n; R) and aeR.

where a= — 1/n +1, b = nIn +1 .

We may describe the graded Lie algebra I as follows. Let 1/ be the n-dimensional vector space of column n-vectors and 1/* be the dual space consisting of row n-vectors. Then I= 17 -1-gI(n; R)+ 1/* under the identification

/0 y) eg _—>ye1/*, i k0 0

(0 0) egi —> E V*, 0

/A 0\ ego —>A—ain egl(n;R). k o al

4. Projective and Conformal Connections

133

Under this identification, the Lie algebra structure in I is given by

[y, V] =0, [, ']=0 ; [U, v]= U v, [, U]= U,

[U,UT.-=UU'—Ui U, where y, We V, , 'E V *, U, U'egl(n; R). Clearly, the element E is now given by — in egI(n; R). With respect to the natural bases el , ... , en of V, e1 , ... , e of V* and ei of gl(n; R), we write the Maurer-Cartan form co of PGL(n; R) as follows:

co =Eal ei +Ea4 el +Ecoi ei . Then the Maurer-Cartan structure equations of PGL(n; R) are given by

d al =

-E Cdk A cok,

d co. I = —E alk A co., — coi A a) i + (5.i Ecok A cok , dcoi = —Ecok A ail. Example 4.2. We describe the n-dimensional Möbius space Sn, first geometrically and then group-theoretically. Let S be the symmetric matrix of order n +2 given by S=

00—1 0 in 0). \—i 0 0

(

Let xeRn+ 2 be a nonzero column vector, considered as a point in the real projective space g +i (R). The quadric Sn in g +,(R) defined by

ixSx=0 is the n-dimensional Möbius space. It is diffeomorphic to the n-sphere 1 in W +1 under the mapping defined in terms of the natural homogeneous coordinate system x°, x1 , ..., xn + 1 of 'F1 (R) by

(y1)2 ± ... _F un2 +1• ) ,

x°=. 1 (1— y n+i),

xn± 1 .--1(1 + yn + 1 ).

Let p be the natural projection from Rn + 2 — {0) onto R +1 (R) and ds 2 be the natural Riemannian metric on g +, (R) defined by

p* (d s 2 ) = 2 [(E xi xi) (E d x i d A—( x dx 1)2 } /(E x i xi) 2. Then the diffeomorphism above is an isometric mapping of the unit sphere onto the Möbius space sr!. Let

2=0(n+1,1)={XeGL(n+2; R); 1 XSX=S}, (a-1 0 0 .20 = { v A +0(n+1, 1); Ae0(n) aeR, eR''} , b a

IV. Affine, Conformal and Projective Transformations

134

where c is written as a row vector. By a simple calculation it can be shown that v=a -1 A•`,

b=(2a) -1

•.

Let

0 21.=.1(1. O0 )e0(n+1, 1); 'eltn},

j,, b

1

Then 2 acts transitively on the Möbius space Sn with 20 as the isotropy subgroup at the point defined by x° = x1 ---- • • • = xn = 0, called the origin of the Möbius space S". The subgroup 21 is the kernel of the linear isotropy representation of 20 at the origin. The graded Lie algebra I= g_ 1 + go +git associated with 2/20 is given by

I=o(n+1,1)=Vegl(n+2; R); 'XS +SX=0), 0 tv 0 9_ 1 = {(0 0 V) 000

(—a 0 0

go--- { - 0 A 0); Aeo(n)}, 00 a

000 gl = {( 1. 0 0 )1, 0 0 where y is a column n-vector, Eego is given by

is a row n-vector and aeR. The element

—1 0 0 E=( 000.

01 As in Example 4.1, we can describe the graded Lie algebra I as I= V+ co(n)+ V* under the identification

(0 'y 0 0 0 v)eg_ i —>yeV, 000 (

(

000 0 0)egi —> eV*, 0

a 00

0 A +90 —>A—ain eco(n). 00 a

0

4. Projective and Conformal Connections

135

Under this identification, the Lie algebra structure in 1 is given by [y, y] =0, [, ']=O, [U, v]=U v, [, U]= U ,

[U, U']= U U' — U' U , [v, where y, y'e V, c, ''e V*, U, U'e co(n). Clearly, the element E is now given by —Le co(n). Using the same bases for V, 11* and gl(n; R) as in Example 4.1, we can write the Maurer-Cartan form co of 0 (n +1, 1) as follows:

co=Ecoi ei +Eco.Vg +Ecoi ej, where (coi) is co(n)-valued. Then the Maurer-Cartan structure equations of 0(n+ 1, 1) are given by

d coi = — E COI A COk , d wf = d wi = —E wk A wl. Let P be a principal 20-bundle over M. Given a g_ 1 -valued 1-form co_ 1 and go-valued 1-form coo , we consider the question whether there exists a natural 91 -valued 1-form col such that co = co_ 1 + coo +col is a Cartan connection in P. There are some obvious conditions which must be imposed on co_ 1 and w0 . Corresponding to conditions (a), (b) and (c) for Cartan connections stated in § 3, we must have the following: (a')

co_ i (A*)= 0 and

wo(A*)= A o for every A ego + gi ,

where Ao is the go-component of A; (b')

W ar (co _ 1 + °k)= (ad a - 1 ) (w_ 1 + coo)

for every a e 20 ,

where ad a-1 is the transformation of g_ 1 + go (= ligi ) induced by ad a': 1—> I; (c') a tangent vector X of P is vertical (i.. e., tangent to a fibre) if co_ i (X)=0. We are now in a position to state a theorem on normal projective and conformal connections. Theorem 4.2. Let .2/ 20 be as in either Example 4.1 or Example 4.2 and P be a principal 20-bundle over a manifold M of dimension n (.. 3 for Ex. 4.2). Given a g_ 1 -valued 1-form w_ 1 =(cd) and a go-valued 1-form coo = (q) on P satisfying conditions (a), (V), (c') and (I)

doii = — E wi A dc,

IV. Affine, Conformal and Projective Transformations

136

there is a unique Cartan connection co= co_ i +0).0 + col = (coi; a4; coi) such that the curvature Q=(0; Cl.;; SO satisfies the following condition:

E 4, =o,

where fl.;= E 1 K ii k i CO k A col.

This unique connection is called a normal projective or conformal connection according as 2/20 is as in Example 4.1 or 4.2. Proof Let co=(coi ; coi; co]) be a Cartan connection with the given (a ; coi). In addition to the first structure equation (I) above, we have

(II)i,

dcoi= —Ecoik A col— coi A coi + 6jEcoo co14S4,

or

(II) c

Ch.0ii= — E COL

A (.0/jf — COL A

Wi — COi A (t)i± 6.1i DOk A COk ± qii ,

according as the connection is projective or conformal, and also dcoi =

(III)

—Ewk A wi.; +Op

Applying exterior differentiation d to (I), making use of (I) and (II) and collecting the terms not involving col and wi, we obtain the first Bianchi identity: QA CO-i

=0,

or equivalently,

14, 1 +K ki u +Ki ik =0. Hence, the condition

EICjii =0 implies also

We shall now prove the uniqueness of a normal Cartan connection. Let Co =(coi ; coi; Q be another Cartan connection with the given (coi ; coi). By conditions (a) and (c) of §3, we can write (T),—(0.,= EA ikC0k,

where the coefficients Aik are functions on P. Denoting the curvature of o by a=(0; and writing

41; ai)

6.= IL E kiki

wk A COI,

we obtain by a straightforward computation using (II) the following relations between Ki m and kiki :

— bi A jk ± 451 A il+ (51.1Akl— Aik (projective), kj k 1 — IC.j k 1 = — 61 AA+ 61 Ail+ M Aik — 6sit Ail+ 6. Akl — (5ijAlk Poi — K.iiki=

(conformal).

137

4. Projective and Conformal Connections

Hence, (i)p

(ii)p (i

)

(ii) c

E (kb,—

azi,„

EA„— E

==(n + 1) (Ala Aik), 1) Ail+ (A11 —

j

(projective)

(n — 2) Ail+ 6iiE A i,,

(conformal)

= 2(n — 1) E Ali .

If both co and are normal Cartan connections, i.e., E Kj ii = E k.i„i = 0, then we see that A u = O and hence co= ro in either the projective or the conformal case. This proves the uniqueness of a normal Cartan connection. To prove the existence, assume first that there is a Cartan connection =(a.)'; c)j; coi) with the given (cd; col). We shall find a suitable system of functions Aik so that ro =(cd; coi; C63) becomes a normal Cartan connection. In the projective case, it is clear from (Op and (ii)p that it suffices to set A1k=

1 4k (n + 1) (n — 1) E

1

n—1

EKiik .

In the conformal case, from (i) and (ii) we see that it suffices to set A :k = j

1 1 6 qc n-2 2(n-1) '

Kiiik

.

To complete the proof of the theorem, we have now only to prove that there is at least one Cartan connection co with the given (cd; 4. Let {U.} be a locally finite open cover of M with a partition of unity { . If co. is a Cartan connection in P IU,, with the given (cd; a4), then E (fx o 7E) COΠa

is a Cartan connection in P with the given (cd; coi), where 1r: P— M is the projection. Hence, the problem is reduced to the case where P is a product bundle. Fixing a cross section a: M—> P, set co3(X)=0 for every vector X tangent to o- (M). If Y is an arbitrary tangent vector of P, we can write uniquely Y Ra (X)+ W, where X is a vector tangent to a(M), a is in 20 and W is a vertical vector. Extend Wto a unique fundamental vector field A* of P with A e!0 go + g. By conditions (a) and (b) for Cartan connections, we have to set

co(Y)— ad(a -1 )(co(X))+ A . This defines the desired (coi).

q.e.d.

IV. Affine, Conformal and Projective Transformations

138

Theorem 4.3. Let P be as in Theorem 4.2 and co = (coi; coi; co) be a normal Cartan connection. Then (1)E A coi =0, or equivalently, Kiki+ K iku+ K iuk= 0, where

Qi=E1-4 / cok A co';

(2)E

Qi = 0, or equivalently, Kiki+ Kku+ Kuk= 0, where Q; =-EA-K iki wk A C01 ;

(3) if Qi= 0, then Q i = 0 provided dim M dim M 4 in the conformal case.

3 in the projective case and

Proof (1) This has been already proved in the proof of Theorem 4.2

whether the connection is normal or not as long as it satisfies (I) of Theorem 4.2. • (2) Apply exterior differentiation d to (MI, and (II), collect those terms involving only w (not (.0.1 and co.) and take the trace. Since c/(21= 0, we obtain the desired result. (3) Similarly, apply exterior differentiation d to (II)p and (II) and collect those terms involving only cd. Since (2.1=0 by assumption, we obtain cos A Q. =0 (in the projective case), co' A Qi — cal A Qi =0

(in the conformal case).

In the projective case, we can conclude immediately that Q. 0 provided n 3. In the conformal case, we rewrite the above identity as 6m1Kikt

— brin Kai

Kira1+ bic Ki m 45IK ikm+ (5I Kam = 0 .

Let i= m, and summing over i we obtain (n— 3) Kikt+ (5ic E K w +

EKik,o.

Summing over j= k, we obtain (2n-4) EKiii =0. From the last two identities we can conclude that K ik i - 0 provided n4. q. e. d. Theorem 4.2 goes back to E. Cartan [2, 4]. Ochiai [3] proved Theorem 4.2 in a very general setting, thus showing that the existence and uniqueness of a normal Cartan connection are related to the vanishing of certain Spencer cohomology groups. Here we followed KobayashiNagano [1] rather closely. For a slightly different approach, see Tanaka [1, 2]. See also Ogiue [1].

5. Frames of Second Order

139

5. Frames of Second Order In preparation for the following section on projective and conformal structures, we shall construct bundles of frames of higher order contact, in particular, of second order contact (see § 8 of Chapter I). Let 'iti be an n-dimensional manifold. If U and I/ are two neighborhoods of the origin 0 of Rn, two mappings f: U-+ M and g: Ti—>M are said to defme the same r-jet at 0 if they have the same partial derivatives up to order r at O. The r-jet given by f is denoted by A(f). If f is a diffeomorphism of a neighborhood of 0 onto an open subset of M, then the r-jet f0 (f) at 0 is called an r-frame at x = f (0). Clearly, a 1-frame is an ordinary linear frame. The set of r-frames of M, denoted by Pr(M), is a principal bundle over M with natural projection it, n(jro (f))= f (0), and with structure group Gr(n) which will be described next. Let Gr(n) be the set of r-frames fo (g) at 0 ERA, where g is a diffeomorphism from a neighborhood of 0 in Rn onto a neighborhood of 0 in W. Then Gr(n) is a group with multiplication defined by the composition of jets, i. e., iro (g) . jr0 (g1) = iro (g o g') . The group Gr(n) acts on Pr(M) on the right by

A (f). j r0 (g) = j'0 ( f o g)

for A (f )e Pr (M)

and fo (g)e Gr(n).

Clearly, 131 (M) is the bundle of linear frames over M with group G1 (n)= GL (n; R) . From now on we shall consider only P1 (M) and P2 (M). If we consider the group 91(n; R) of affine transformations of Rn as a principal bundle over R" = 91(n; R)/GL(n; R) with structure group GL(n; R), we have a natural bundle isomorphism between 91(n; R) and the bundle (Rn) of linear frames over Rn:

Pi

91 (n ; R) 4---

I

Pi (Rn) 1.

Under this isomorphism, the identity e of 21(n; R) corresponds to ,Po. (id), where id denotes the identity transformation of W. We shall therefore denote A (id) by e. The tangent space off.' (R") at e will be identified with that of 91(n; R) at e, that is, with the Lie algebra

a(n; R)=Rn+gi(n; R) of 9.1(n; R).

140

IV. Affine, Conformal and Projective Transformations

We shall now define a 1-form on P2 (M) with values in a (n; R). First, we observe that j1(f)—)j4(f) defines a bundle homomorphism 132 (M)— P1 (M). Let X be a vector tangent to P2 (M) at A(f) and X' be the image of X under the homomorphism P2 (M)—)131 (M). Then X' is a vector tangent to P1 (M) at j131 (f). Since f is a diffeomorphism of a neighborhood of the origin 0 of Rn onto a neighborhood off(0)eM, it induces a diffeomorphism of a neighborhood of eeP1 (IV) onto a neighborhood of j4(f)eP 1 (M). The latter induces an isomorphism of the tangent space a (n; R)= 7; (P1 (Rn)) onto the tangent space of P1 (M) at j4(f ); this isomorphism will be denoted by! and easily seen to depend only on A(f). The canonical form 0 on P2 (M) is an a (n; R)-valued 1-form defined by 0 (X) = f -I (X') .

The construction generalizes that of the canonical form of 131 (M). We define the adjoint action ad of G 2 (n) on a (n; R) as follows. Let A (g)e G 2 (n) and A(DeP 1 (Rn). The mapping of a neighborhood of e E P1 (W) onto a neighborhood of e E P1 (W) defined by A (f )—)ii(1) (g of o g— 1 ) induces a linear isomorphism of the tangent space a (n; R)= 7e (P1 (m) onto itself. This linear automorphism of a (n; R) depends only on j (g) and will be denoted by ad (j (g)). Since G 2 (n) acts on P2 (M) on the right, every element A of the Lie algebra g2 (n) of G2 (n) induces a vector field A* on P2 (M), called the fundamental vector field corresponding to A Proposition 5.1. Let 0 be the canonical form on P2 (M) defined above. Then

(1) 0(A*)= A' for A E g2 (n), where A' e gl (n)= gl (n ; R) is the image of A under the. homomorphism g2 (n)—> g1 (n); (2) (R.)* 9=: ad (a -1 )0 for a e G 2 (n). The proof is straightforward. , We shall now express the canonical form of P2 (M) in terms of the local coordinate system of P2 (M) which arises in a natural way from a local coordinate system of M. For this purpose we may restrict ourselves to the case M = W. Let e1 , ... , en be the natural basis for Rn and (x1 , ... , xn) be the natural coordinate system in W. Each 2-frame u of Rn has a unique polynomial representation u=j(f) of the form f (x)=E(u s +Eu 1 xi +4Eulk xi xk) ei ,

where x =E x1 ei and 141, -=uli . We take (u1 ; 14; ulk) as the natural coordinate system in P2 (Rn). Restricting (i4; f.4k) to G 2 (n) we obtain the

6. Projective and Conformal Structures

141.

natural coordinate system in G 2 (n), which will be denoted by (si; The action of G 2 (n) on P2 (W) is then given by

uip sy ; E upi syk +E uqf r

(ui ; uS; ulk)(sii ; 4k)=(ui ;

In particular, the multiplication in G 2 (n) is given by

Similarly, we can introduce the natural coordinate system (ui ; u. ) in P1 (W) and the natural coordinate system (.s.i) in G 1 (n) so that the homomorphisms P2 (Rn) —> P1 (Rn) and G 2 (n)—> (n) are given by (zit ; uti)

(ui ;

and

(si; sik) > —

respectively. Let (Ei ; El} be the basis for a (n; R) defined by

Ei =

la

,

El= (i/

and set

a di)e

0=E0t E1 -FE0jEj. From the definition of the canonical form 0 and the action of G2 (n) on

p2 (Ra) expressed in terms of the natural coordinate systems, we obtain

the following formulae: dui =Eulei,

duti —Eulei+Euihi 0h. Let (vi) be the inverse matrix of (i4). Then O1 —>v duk,

=Evik From these formulae we obtain the following structure equation: Proposition 5.2. Let 0 = (0 i ; O. ) be the canonical form on P 2 (M). Then

d01 =—E0ik nOk. For the canonical form on P' (M) for r >2 and its structure equation, see Kobayashi [8]. This set-up of jets, frames of higher order contact, ete. is due to Ehresmann [3]. 6. Projective and Conformal Structures Let 2/20 be as in Example 4.1 or 4.2. We can consider .2 0 as a subgroup of the group G2 (n)-- ((c ) ; aiik)) defined in §5 as follows. Let o denote the origin of the homogeneous space 2/2 0 and consider each element g of

142

IV. Affine, Conformal and Projective Transformations

20 as a transformation of 2/2 0 leaving the origin o fixed. It can be easily verified that j(23 (g)=id if and only if g = id, that is, every element of 20 is determined by its partial derivatives of order 1 and 2 at the origin o. Hence, 20 is isomorphic to the group of 2-jets (A(g); g ego). Choosing a basis for g_ i , identify g_ i with V= Rn as in Example 4.1 or 4.2. Then the mapping ' 2—> 2/20 Rn = 1 e'> gives a diffeomorphism from a neighborhood of 0 eRn onto a neighborhood of o E 2/20 and defines a local coordinate system around o ewe ° . With respect to this coordinate system, each 2-jet A(g) is an element of the group G 2 (n). Hence, 20 can be considered as a subgroup of G 2 (n). An explicit description of 2 0 as a subgroup of G2 (n) is not without interest although it is not essential in the subsequent discussion. Let N2 (n)= MO) denote the kernel of the natural homomorphism G 2 (n)—> On) which sends (ai; ciik) into (ai). In view of the diagram of exact sequences: —> (n)—> G 2 (n)—> (n) —> 1

U U U 0—> 21 —> 20 —> 20/21 —> 1, we shall describe 2 0/21 as a subgroup of G1 (n) and 21 as a subgroup of N2 (n). First, let weo be as in Example 4.1. Then 20/21 = G1 (n)= GL(n; R) and 21 = {(dik); Next, let 2/20 be as in Example 4.2. Then 2 0/21 = CO (n) and 21 =

( (aik); a.ik=

±

J

i) •

Thus 21 coincides with the first prolongation of the Lie algebra co(n) (see Example 2.6 of Chapter I). Let M be a manifold of dimension n and P 2 (M) be the bundle of 2frames over M with structure group G 2 (n) (see § 5). A principal subbundle P of P2 (M) with structure group 20 (c G2 (n)) is called a projective structure or a conformal structure on M according as 2/20 is as in Example 4.1 or 4.2. (In Example 2.6 of Chapter I, a CO(n)-structure on M was called a conformal structure on M. The two definitions are equivalent in the following sense. If P is a conformal structure as a subbundle of P2 (M), then P/21 is a subbundle of P 1 (M).- L(M) with group 20/21 = CO(n) in a natural manner and hence is a CO(n)-structure. Conversely, if Q is a CO (n)-structure on M, then its first prolongation Q1 (see §5 of Chapter I) is a conformal structure as a subbundle of P 2 (M). This gives a one-to-one correspondence between the conformal structures P c P2 (NI) and the CO (n)-structures Q c P 1 (M). Since we shall not use this fact, the

6. Projective and Conformal Structures

143

proof is left to the reader. But a projective structure P cannot be defined as a subbundle of P1 (M)= L(M) since P/21 is 131 (M) itself and the first prolongation of P1 (M) is precisely P2 (M).) Now, let P c P2 (M) be a projective or conformal structure on M and (cot ; co be the restriction to P of the canonical form (O'; Of,) of P 2 (M) (see § 5). By Propositions 5.1 and 5.2, (coe ; cf4) satisfies the assumptions (a'), (b), (c') and (I) of Theorem 4.2. We shall therefore call (coi; 04) the canonical form of P. By Theorem 4.2, there is a unique normal projective or conformal connection (co1 ; co. ; co) provided n 3. Let P and P' be projective (resp. conformal) structures on manifolds M and M', respectively. A (local) diffeomorphism f of M into M' induces a local isomorphism f* of the bundle P 2 (M) of 2-frames of M into the bundle P2 (M') of 2-frames of M'. If f* sends P into P', then f is called a (local) projective (resp. conformal) isomorphism of M into M'. If M = M' and P = P', then a projective (resp. conformal) isomorphism f is called a projective (resp. conformal) transformation or automorphism. This definition is completely analogous to that of an automorphism of a G-structure given in § 1 of Chapter I. An infinitesimal projective or conformal transformation can be defined in the same way as an infinitesimal automorphism of a G-structure. Assume that n 3 so that the normal projective or conformal connection is unique. Then, for each automorphism f of P, f (restricted to P) preserves the normal connection co =(w'; coii ; coi). From Theorem 3.1 we obtain (Kobayashi [2]).

Theorem 6.1. If P is a projective or conformal structure on a manifold M of dimension n 3, then the group Qt of projective or conformal transformations is a Lie transformation group of dimension dim P (= n2 +3 n in the projective case and = + 1)(n +2) in the conformal case). In order to determine the cases where dim 91 , dim P, we consider some examples. Let 2/20 =I(R) be as in Example 4.1, where .2 = PGL(n; R). The principal bundle .2 over 2/20 with group 20 can be identified with a projective structure on 2/20 in a natural manner. To describe this identification, let o denote the origin (i.e., the coset 2 0) of the homogeneous space 2/20 . Since each fe .2 is a transformation of 2/20 and a neighborhood of o in 2/20 is identified with a neighborhood of 0 in Rn in a natural way, the 2-jet j (f) can be considered as a 2-frame of 2/20 The set of all 2-frames thus obtained defines a projective structure atf(o). on 2/20 which can be identified with the bundle L over 2/20 . Then the Maurer-Cartan form co =(co1 ; coi ; ao of .2 defined in Example 4.1 defines the normal projective connection of this projective structure. The Maurer-Cartan structure equations of .2 show that the connection has

144

IV. Affine, Conformal and Projective Transformations

no curvature. Clearly, Q is the group of projective transformations of this projective structure. We consider now the universal covering space sr: of F(R). Since the I(R) is a local diffeomorphism, the natural covering projection projective structure on I(R) described above induces a projective structure on Sn. We shall give a group-theoretic description of this natural projective structure on S". Let .2 = GL(n + 1; R)/R+, where R + denotes the normal subgroup of GL(n +1; R) consisting of elements cdn+i with a> O. It is a group with two components which is locally isomorphic to PLG(n; R). (PLG(n; R)= GL(n + 1; R)/R*, where R* is the normal subgroup consisting of elements a In +1 with a+ 0.) Let

20=1 (A a eGL(n+1; R); a>01/R+, where A E GL(n ; R) and is a row n-vector. Then Sn =2/20 . The principal bundle 2 over 2/20 with group 20 is the desired projective structure on Sn which is locally isomorphic to the natural projective structure on pn (R) under the covering projection Sn—)1(R). The group of projective transformations of this projective structure is Q. As the third example, let 2/20 sr, be as in Example 4.2, where 2= O (n +1, 1). Then the principal bundle 2 over 2/20 can be naturally identified with a conformal structure on 2/20 . Again the Maurer-Cartan form co= (a); coi; co) of Q defined in Example 4.2 defines the normal conformal connection of this conformal structure. As we can see from the Maurer-Cartan structure equations of 2, the connection has no curvature. The group .2 is precisely the group of conformal transformations of this conformal structure. Theorem 6.2. In Theorem 6.1, assume dim 91= dim P. In the projective case, P is the natural projective structure on either Pn (R) or its universal covering space Sn as explained above. In the conformal case, P is the natural conformal structure on Sn described above.

We shall only indicate the main idea of the proof. We consider the projective case, the conformal case being similar. Since each 1-parameter group of projective transformations of M lifts to al-parameter group of projective transformations of the universal covering space JÇ the group of projective transformations of /i/-1 has the maximum dimension dim P. We shall first assume that M is simply connected. Choose.a point u0 of P. We know (Theorem 3.2 of Chapter I) that the mapping fe91—>fit (uo)eP is injective and the image is a closed submanifold of P which can be identified with 91 (as a differentiable manifold). Since dim 91= dim P by assumption, the identity component of 91 can be identified with one of the components of P under the mapping 91—) P defined above. (Since P ,

7. Projective and Conformal Equivalences

145

has two connected components, 91 can be identified with P if 91 is, not connected.) Let xo be the base point of uo in M and 910 be the isotropy subgroup of 91 at xo . Since 91 is fibre-transitive on P, it is transitive on M so that M = 91/91g . Let co = (d; cc4; wi) be the normal projective connection of P. Since the forms (a); q; co) are invariant by 91, restricted to 91c P they are Maurer-Cartan forms of the group 91. Since the curvature of co vanishes by Theorem 3.2, the structure equations (I), (II)p , (III) of § 5 are nothing but the Maurer-Cartan structure equations of the group 91. From these structure equations we see that 91 is locally isomorphic to 2 = GL(n+ 1; R)/R +. The identity component of 210 coincides with the identity component of the structure group 20 . Since both 91/210 and 2/20 are simply connected, the standard argument proves that the identity component of 21 and the identity component of 2 are isomorphic to each other not only as groups but also as bundles over M. 91/910 and Sn =2/20 respectively. Then it follows that the bundle P over M is isomorphic to the bundle 2 over Sn. If M is not simply connected, then M= Sn/T, where T is a discrete subgroup of 2 = GL(n +1; R)/R + which commutes elementwise with the identity component GL E (n + 1; R)/R + of 2 as in the proof of Theorem 3.1 of Chapter II. (Here GL + (n +1; R) denotes the identity component of GL(n +1; R), which consists of matrices with positive determinant.) Then we see that 11 consists of two elements represented by + i n+i , and that P is the natural projective structure on M = pn (R).

7. Projective and Conformal Equivalences We shall first explain projective equivalence of affine connections. Let I= g_ 1 + go + git = Y+ g! (n; R)+ 17* be as in Example 4.1. Let P1 (M) (= L(M)) be the bundle of linear frames over M and 0 =WI be the canonical form of P1 (M), viewed as a g_ 1 -valued 1-form as well as a Y-valued 1-form. Two torsionfree affine connections of M defined by go-valued 1-forms co.(coi) and co' =((o'i ) are said to be projectively equivalent if there exists a grvalued function p = (pi) on P1 (M) such that co' — co= [0, p].

This formulation is due to Tanaka [4] (see also Kobayashi-Ochiai [1]). Note that the left hand side takes values in go and the right hand side in [g_ 1 , gi ] c go . If we consider co and co' as gl(n; R)-valued forms, 0 as a Y-valued form (i.e., a form whose values are column n-vectors) and p as a Y*-valued function (i.e., a function whose values are row n-vectors), then the relation above may be rewritten as follows:

co' —0)=-Op+(p0)1„,

146

IV. Affine, Conformal and Projective Transformations

or more explicitly 19 '

P (E °I‘ P k)

If we set then the relation reads as follows:

ri k =61pi +5ipk . Similarly, we define conformal equivalence of affine connections. Let I= g_ 1 + go + git = V+ co (n)+ V* be as in Example 4.2. Let Qc P1 (M) be a CO (n)-structure over M and O = (0) be the canonical form on Q. Two torsionfree connections in Q defined by go-valued 1-forms co = and co' =(ctii i) on Q are said to be conformally equivalent if there exists a g1 -valued function p =(p i) on Q such that

cd —0)=M A, or, in terms of matrix and vector notations, al —0)=0 p—'pl0+(p19) or more explicitly,

— coi = 19' p — 9i p,+ (E O k p If we set

wJ -wJ =rJk O, then the relation reads as follows: ilk= 6 1P biPi+ Although two torsionfree connections in P1 (M) are not necessarily projectively equivalent, two torsionfree connections c o = (0)i) and co'=(ali i) in the CO(n)-structure Q are always conformally equivalent to each other. This difference comes from the fact that while' g 1 is the first prolongation of go in the conformal case, g1 is strictly smaller than the first prolongation of go in the projective case. In fact, at each point of Q, (ri k) defines an element of the first prolongation of go since, for each fixed k, the matrix (riik)0.0i , ...,„ is in co(n) and (r.k) is symmetric in j and k. In Example 2.6 of Chapter I, we proved that each element (ilk) of the first prolongation of co (n) determines a unique vector (pi) such that rt., k = 61 biPi+ (5; Pk' In order to relate the notion of projective equivalence to that of projective structure, we prove the following

7. Projective and Conformal Equivalences

147

Proposition 7.1. Let P2 (M) be the bundle of 2-frames over an n-dimensional manifold M with structure group G 2 (n). Consider G 1 (n)=--GL(n; R) as the subgroup of G 2 (n) consisting of elements (si; s) with sik =0 in terms of the natural coordinate system introduced in § 5. Let 2/ 20 . g(R) be as in Example 4.1 and consider .20 as a subgroup of G 2 (n) as in §6 so that (n) 0 c G 2 (n). Then (1) The cross sections M— P2 (M)/G 1 (n) are in one-to-one correspondence with the torsionfree affine connections of M. (2) The cross sections M—> P2 (M)/ 130 are in one-to-one correspondence with the projective structures of M.

4) be the natural local coordinate system in (M) induced from a local coordinate system x 1 , , xn of M as in § 5. We introduce a local coordinate system (zi; zi k) in P2 (M)/G1 (n) in such a way that the natural projection P 2 (M)—> P2 (M)/G 1 (n) is given by the equations zi= = uipq Proof (1) Let (u';

fic

where (v. .:(0-1 )

Then a cross section functions

(the inverse matrix).

: M —> (M)/ (n) is given locally by a set of risk (xl , , xn) with rijk -Tikj •

If we consider the action of the group G2 (n) on the fibre G 2 (n)/G 1 (n), then we see that the functions ilk behave under the coordinate changes as Christoffel's symbols should. This proves (1). (2) Since the reductions of the structure group G 2 (n) to 20 are in one-to-one correspondence with the cross sections M —> P2 (M)/ 2 0 , (2) is evident. q. e.d. Let 0 =(13 : • 0) be the canonical form on P 2 (M) defined in §5 and y: 131 (M) c--> (M) be the reduction of the structure group G 2 (n) to G 1 (n) corresponding to a cross section : M — > P 2 (M)/ (n). Then (y* (01) is the canonical form of 131 (M) and (y*(60i)) is the connection form corresponding to F. This follows easily from the proof of Proposition 7.1 and the expression of 0 given in §5 in terms of the coordinate system (u`; Every torsionfree affine connection F: M—* P2 (M)/G 1 (n), composed with the natural mapping P2 (M)/ G 1 (n)—> P 2 (M)/20 , gives a projective structure M —> P 2 (M)/2 0 . A torsionfree connection r is said to belong to a projective structure P if it induces P in the way described above. • Proposition 7.2. Two torsionfree affine connections of a manifold M are projectively equivalent if and only if they belong to the same projective structure.

148

IV. Milne, Conformal and Projective Transformations

Proof Let I' and I" be two cross sections M —) 132 (M)/G 1 (n), i.e., torsionfree affine connections and let co and co' be the corresponding

connection forms on 131 (M). A straightforward calculation shows that co and co' are projectively equivalent if and only if there is a 1-form p= p i dx' on M such that Gcg — Elk = (51 Pk -F (51P], where Tiki and Fik are Christoffel's symbols for co and co' with respect to a local coordinate system , xn. This in turn is equivalent to the condition that I" and I" induce the same cross section M—> P2 (M)/2 0 , because the kernel 21 of the homomorphism Q 0 —> On) consists of elements (ah) of the form a.k = pk + 5 p (see § 6). q. ed. Given a torsionfree affine connection on M, let 91(M) be the group of affine transformations of M and 13(M) be the group of projective transformations, i. e., automorphisms of the induced projective structure. Proposition 7.2 implies that a transformation of M is a projective transformation if and only if it transforms the given connection into a projectively equivalent affine connection. Evidently, we have the inclusion 91(M) c 43 (M). There are cases where these groups have the same identity component. We quote only the following result of Nagano [8]. .

Theorem 7.3. Let M be a complete Riemannian manifold with parallel Ricci tensor. Then the largest connected group 130 (W) of projective transformations of M coincides with the largest connected group 9.1° (M) of affine transformations of M unless M is a space of positive constant curvature.

For other related results, see Couty [2, 3], Ishihara [2, 3], Solodovnikov [1], Tanaka [2], Tashiro [3], Yano [3], Yano-Nagano [2], and references therein. Given a CO (n)-structure Q c P1 (M), let P c P2 (M) be the corresponding conformal structure on M. Then a transformation of M is an automorphism of the CO (n)-structure Q if and only if it is an automorphism of the conformal structure P. Such a transformation is called a conformal transformation of M. If M is a Riemannian manifold, then a transformation of M is a conformal transformation with -respect to the naturally induced CO (n)-structure if and only if it sends the metric into a conformally equivalent metric (cf. Example 2.6 of Chapter I). Evidently, the group E (M) of conformal transformations of M contains the group 3(M) of isometries of M. We quote only two results on the relationship between E (M) and 3(M).

Theorem 7.4. Let M be a complete Riemannian manifold with parallel Ricci tensor. Then the largest connected group E° (M) of conformal trans-

7. Projective and Conformal Equivalences

149

formations of M coincides with the largest connected group 3° (M) of isometries of M unless M is isometric to a simply connected space of positive constant curvature ( i.e., a sphere).

This result of Nagano [1] extends the result of Yano -Nagano [1] on Einstein manifolds. The following theorem was proved by Obata [8]. See Lelong-Ferrand [4] for related results. 'theorem 7.5. Let M be a compact Riemannian manifold with constant scalar curvature. Then E° (M) . 3° (M) unless M is isometric to a simply connected space of positive constant curvature.

For other related results, see Goldberg-Kobayashi [1], Kulkarni [1], Ledger-Obata [1], Lichnerowicz [1], Obata [3-7, 9], Suyama-Tsukamoto [1], Tanaka [1], Tashiro [2], Tashiro-Miyashita [1, 2], WeberGoldberg [1], Yano [3, 8], Yano-Nagano [2], Yano-Obata [1], and references therein. A list of papers which had given partial results toward Theorem 7.5 can be found in Yano-Obata [1].

Appendices

1. Reductions of 1-Forms and Closed 2-Forms

For the discussion of contact structures and symplectic structures, it is important to know the simplest possible expressions for 1-forms and closed 2-forms. Let a) be a 1-form defined on a manifold M of dimension n. We set d2k) = CO A ••• A da) (k times), (2k+I)= condom—

Ada)

(da): k times).

Then cdP ) is a form of degree p. We say that a 1-form a) is of rank r at a point o if co(r) +0 but aP+ 1) =0 at o. Then we have Theorem 1. If a 1-form a) is of rank p in a neighborhood of a point o, then there exists a local coordinate system x1 , xn around o such that (0=x ' d x 2 + x 3 d x4+ ± x 2k-1 d x 2k for p = 2 k, w=x dx2+x3 dx4+• +x2k-1 dx2k+dx2k+i

for p=2k+1.

Proof We first prove

Lemma 1. If co is of rank p, then there exist a neighborhood Un of o, Un—> UP with dim UP =p and a 1-form Co on UP such that a fibring ir* (&)=w. In other words, with a suitable choice of local coordinate system, co depends only on the first p coordinate functions. Proof of Lemma 1. At each point we consider the space S of tangent vectors X such that ix a)=0

and

tx da)=0,

where ix denotes the interior product. It is straightforward to verify that if X and Y are vector fields which belong to S at each point, then [X, Y] belongs to S at each point. We shall show that dim S=n—p. (That will mean that S defines an involutive distribution of dimension n — p.) Set p =2k or p =2k +1 according as p is even or odd. Then at each point,

1. Reductions

of 1-Forms and Closed 2-Forms

151

da) is a skew-symmetric bilinear form of rank 2k on the tangent space. If we denote by S' the space of tangent vectors X such that ix da)=0,

then dim S' = n —2k and S c S'. If p= 2k, then S = S'. In fact, if S *S' and YES' is an element not in S, then co(Y)* 0. If X1 , ... , X2 k are tangent vectors (not in S') such that V

1

Ji L 2k1 = 010) A

••• A

d(o)(X l , ..., X 2 0#0,

then , co(2k+1)11, ki, A I ,

...,

X2k)=00 A dWA ••• A dC0)(Y, XI , ..., X20 =COOT a (2k) (XI, ...,X20#0,

in contradiction to the assumption that cook" = 0. If p= 2 k+ 1, then dim S' =1+ dim S. In fact, since S = {X e S'; a)(X)= 0), we have dim S S dim S' 1+ dim S. It suffices therefore to prove S* S'. Since w(2k + 1) *0, there exists a (2k + 1)-dimensional subspace T of the tangent space at each point such that the restriction of aP k + 1) to T is nonzero. In other words, for any basis X 11 ...,X2k +1 of T, we have a 2k + 1)(XI , ..., X 2k+1 )# O. But, if S=S', then dim(Sn T)dimS+dim T—n=n-2k+2k+l—n=1. By taking a basis X i , ... , X2k +i of T in such a way that X1 ES nT, we obtain (0 (2k+l)ty y k.exi, ...m2, ••• 1 X2k + i)=(.0 A dw A ••• A dco)(X i , X2, ••., X2k + i) =0

since ixi co =0 and i1 da)= O. This is obviously a contradiction. We have shown that S defines an involutive distribution of dimension n — p. Consider the maximal integral submanifolds defined by the distribution. They give a fibring of a suitable neighborhood Un of o whose fibres are of dimension n — p. We denote this fibring of Un by n: U's—> U. From the definition of S it is clear that co can be projected onto UP, that is, there exists a (unique) 1-form 6 on UP such that n* (6' )= co. This completes the proof of Lemma 1. Lemma 2. If a) is of rank n, then there exists a function f such that df *0 and df A dn -1) =0 in a neighborhood of o. Proof of Lemma 2. We write

an - 1) .E (— oi ai dX 1 A •• • A dx i A • •• A dxn. i

152

Appendices

Then the equation df A co(n-1) = 0 reduces to f

a xi

.o.

This partial differential equation admits (infinitely many) solutions f such that df *0 in a neighborhood of o.

Lemma 3. If co is of rank p, then there exists a function f such that df *0 and dfn Proof of Lemma 3. This is immediate from Lemmas I and 2.

It is also evident that in Lemma 3 we can choose f in such a way that Un P given in Lemma 1. it is constant on each fibre in the fibering We say in general that a form co on M depends on parameters ti , , tm if it is a form on M x N, where N is the space of parameters such that its restriction to (p) x N vanishes for every point p of M, i.e., such that it does not involve d t 1 , , d t„, when expressed in a local coordinate system. The exterior derivative dco of such a form co on M , tm which depends on the parameters t1 , , tm is taken considering as constants. In other words, we take the ordinary exterior derivative of co as a form on M xN and then delete the terms containing dt i ,...,dt m . Looking at the proofs of Lemmas 1 and 2, it is clear that Lemma 3 may be generalized to the case where co depends on parameters. We have

Lemma 4. If, in Lemma 3, co depends on m parameters t1 , , t„„ then we can find a function f which depends differentiably on , tm . Making use of these lemmas, we shall now prove the theorem. We denote by y' the function f obtained in Lemma 3 and try to find a new function f such that dfn d yl * 0

and

dfn d y l co(P - 3) =0.

Choose a local coordinate system y', u 2 , , un around o; since d y l *0, such a local coordinate system exists. We write w=hdy l +q),

where q) does not contain dyi. We consider a neighborhood of o of the form C.J1 X U2 such that y' is a coordinate system in LI, and u2, , Un is a coordinate system in 112 in a natural manner. Then q) may be considered as a 1-form on U2 which depends on the parameter ;I'. We shall denote (p by ep when we consider it as a form U2 which depends on the parameter y'. By direct calculation we obtain dy l A coo = d

ep' (J) for every j 1.

1. Reductions of 1-Forms and Closed 2-Forms

153

Setting j = p —1, we see that ip (P - 1) =0. By direct calculation we obtain also = (k a d + de()) A 2) , where p = 2k or p= 2 k + 1 according as p is even or odd and a is a 1-form such that dco =a A dy l +chp. Since co(P) *0, we have ep"(P- 2) 40. Hence ip is of rank p — 2. Applying Lemma 4 to rp, we obtain a function f on U2 such that df *0 and df A 45( =0. We can extend f to a function on Ui x U2 in a natural manner; the extended function will be denoted also by f Then dyl A df * 0 and d df A CP ." 3) = d yl A df A rp (P - 3) = 0. Setting y2 =f, we may write d y2 + 0

d

and dyl A dy2 Aco(P -3) =0.

Continuing in this way we obtain functions d

A •••

and d

A

...

A

d

yic such that

Ad A co(P- 2k + 1) =0

According as p = 2 k or p = 2 k + 1, we have dy l d

•••

dy k

co=0

Ady k Adco=0

if p=2k, if p=2k+1.

If p = 2 k, we have therefore

co=z i d yl ± z 2 d y 2 where z1 , that d 3/1 A

±zk d y k,

zk are functions. Since OP = d co A .•• A d y k A dz 1 A ... A d z k 4 0. Setting X 1 =Z 1 ,

x2=y1,...,x 21c-1 = Z k

...

A

d co 4 0, it follows

X 2 k =y k

and defining x 2 " 1 , , xn suitably, we obtain a local coordinate system , xi' with the desired property. If p = 2 k +1, we take a local coory k , uk+1 , ... , un since d ••• A d yk + 0, such a dinate system y l , coordinate system exists. We then write ;

dy 1 +.•.+hk dy k +0, where is a 1-form which does not involve dy l , , dy k. As we have explained before Lemma 4, ik may be considered as a 1-form on the space tin which depends on the parameter , yk and will be of uk + 1 , un. Hence di k denoted by when it is considered as a form in uk + 1 , is obtained from dtp deleting the terms involving dy i , ...,dy k. Since 0= dy l A ... A dy k A dco=dy l A ••• A dy k Ad tk, it follows that d =0. By y k, uk + 1 un such Poincar6 lemma, there exists a function g of y l ,

Appendices

154

that

dg=ili+g l dy 1 +•••+gk dy k, where gi = a ea yf. Hence we have

co=(hi —g1 ) dy 1 +•••+(hk —gk)dy k +dg. x 2k = y k, x 2k+1 = g_ , then x 2k--1 = hk _ If we set xl = —gi, x2 =y1 , dx 1 A dX 2 A ••• dx 2k + 1 +0 since a 2k + 1) +0. Choosing x 2k + 2 , ...,xn suitably, we obtain a local coordinate system with the desired property. q. e.d.

Let 0 be a 2-form defined on a manifold M of dimension n. We say that 0 is of rank 2p at a point o if Q =QA A (p times) is nonzero but 02 + 1 =0 at o. We say that Q is of maximal rank if it is of rank n. As an application of Theorem 3.1, we prove Theorem 2. If a closed 2-form Q is of rank 2p in a neighborhood of a point o, then there exists a local coordinate system x',...,xn around o such that

C2=dx 1 dx 2 +...+dx 2 P -1 Adx.

Proof. Since Q is closed, by Poincaré lemma we have Q=dco, where co is a 1-form defined in a neighborhood of o. Then co is of rank either 2p

or 2p +1. By Theorem 3.1 there exists a local coordinate system x 1 , xn around o such that either w=xidx 2 +...+x 2 P-1 dx2 P Or

co=xidx2 +.-±x 2 P-1 dx 2 P+dx2 P+1 . Then it is evident that Q has a desired expression.

q.e.d.

Theorem 3.1 is due to Frobenius and Darboux. We followed the presentation of E. Cartan in [9] to which we refer the reader for relevant references. See also Arens [2]. 2. Some Integral Formulas We prove first the following integral formula of Yano [6] (see also Yano and Bochner [1]). Theorem I. Let M be a compact, orientable Riemannian manifold with Riemannian connection V and Ricci tensor S. Then, for every vector field X on M, we have S (S(X, X)+ trace(Ax A x )—(div X) 2 ) d v =0 ,

2. Some Integral Formulas

155

where A x is the field of linear endomorphisms defined by A x Y = —Vy X, div X is the divergence of X and dv denotes the volume element of M. Proof The proof is in terms of local coordinates. It suffices to express

the integrand as the divergence of a vector field. Let e be the components of X with respect to a local coordinate system x', , xi'. Then

div (A x X)= —E

-E

•+

-E (vi

• e+ RIO

i,j

=

e

i,j

e•

•v +Vj e • Vi ±Vj • VI V)

and •

=E vi e • V+

• NW).

From these two equalities, it is clear that the integrand in Theorem 1 is equal to — div (A x X) — div ((div X) X). q. e. d. In terms of a local coordinate system, the formula in Theorem 1 reads as follows: E (Ri; e +Vi e • V —V; • Vi 0=0. f,

From Theorem 1, we obtain immediately the following three formulas: Corollary. Hith the same notations as in Theorem 1, we have

(1) J {S(X, X)+ trace(Ax Ax)-Ei trace ((Ax — A x)2)— (div X) 2} dv=0;

(2) f IS(X, X)— trace (A x .14x)—i trace ((Ax + A)2)— (div X) 2 } dv=0; f IS(X, X) — trace (A x x ) (3)

2 (div X) .02) — n — 2 (div X)2} dv=0. +4 trace (( A x +14x — — We shall now prove a Kdhlerian analog of Theorem 1. Theorem 2. Let M be a compact Kahler manifold with Ricci tensor S.

For any complex vector fields X and Y of type (1,0), we have

{S

(x, r7)+ trace(A;:. 4)— (div X)(div V)) dv=0 ,

Appendices

156

where A x" is the field of linear transformations sending a vector field type (0, 1) into the vector field —V E X of type (1, 0), i.e.,

of

—VE X. Before we start the proof, we remark that A sends a vector field of type (1,0) into a vector field of type (0, 1) so that A x" • diiT sends a vector field of type (1,0) into a vector field of the same type. We note also that, by setting X = y in the formula above, we obtain a Kahlerian analog of the formula in Theorem 1. Proof In terms of a local coordinate system zl, x=

ta

a a za

and

aa a zet

Y=Etr

zn, let .

Then the components of Ai; are given by — The components of the vector field A(Y)= — Vy X of type (1,0) are given by —E ?. A calculation similar to the one in the proof of Theorem 1 yields

—E vcc (vpdg TIP). — (V, va • TIP +

Πill +



)

and

E yva•iP)=E(vp

Vg By adding these two equalities and integrating the resulting equality over M, we obtain E (Rap OE -r-711 + Vij • VŒ F —VŒ Œ • Vp ill) dv =0 , ± Vac

a

Af

which is precisely the formula in Theorem 2.

q.e.d.

Let X be a vector field on a Riemannian manifold M. Let be the 1-form corresponding to X under the duality defined by the Riemannian metric. If we denote by A the Laplacian, then A c is a 1-form. We denote by A X the vector field corresponding to A We state Theorem 3. Let M be a compact orientable Riemannian manifold with Ricci tensor S. Then, for any vector fields X and Y on M, we have

(— (A X, Y)+ S(X, Y)+ (VX, VY)) dv=0. At Proof Let c and ni be the components of X and Y with respect to a local coordinate system x', , xn. Combining

E co .10=E vi

ni +v' Nini)

with the formula for the Laplacian (see Appendix 3), we obtain

Evvi • to= —DA V. tli+ERuVri i +EVT*Virli. This, integrated over M, yields the desired formula

q.e.d.

157

3. Laplacians in Local Coordinates

Let X be a complex vector field of type (1,0) on a Kdhler manifold Ai

a

—, then a za = E dr, where = E v3.) Let A" =6" d" + d" 6" be the d"-Laplacian. Then A" c is a (0, 1)-form. We denote by 4" X the corresponding vector field of type (1, 0). As a Kdhlerian analog of Theorem 3, we obtain and let

be the corresponding (0, 1)-form. (If X =

Theorem 4. Let M be a compact Kahler manifold with Ricci tensor S. For any complex vector fields X and Y of type (1, 0), we have (— (A" X, Y)+ S(X, Tr) + (v" x, V" Y)) dv = 0. Proof. From

Eva(va e • t70)=-E(va va e3 • Fil+ va v3 • vcc and from the formula for the d"-Laplacian (see Appendix 3), we obtain the desired formula in the same way as in the proof of Theorem 3. q.e.d. Expressed in terms of local coordinates, we obtain

{—E(z1"Dpie-FERzpŒijfl+EV0 Œ.Vs zt} dv=0. 3. Laplacians in Local Coordinates Let co be a p-form on a Riemannian manifold M. Let

1 C4)=-F EW1112...ip

dx il A dxi2

A A

dx iP

be a local expression for co. Then 1 dco=

(p+1)! dx A dxfl A ...

••• lp

A

dx fP,

and co =

1 (p-1)!

. 12 ...1„. dx f2 A

dxiP.

In particular, let co be a 1-form so that

co=Ecoi dx i. Then

da)=1E(Vi coi —Vi co1)dx 1 ndx1, dco= —EVÎ (V i coi —Vi coi)d bco= dbco=

—DOW dxj.

4

. 4

158

Appendices

Hence, do=bdco+dbco =E( — NW' wi+ViVid —V j Vi coi)d xi =E( —DI V coi + Rt., col) d x-I.

For a similar expression for a p-form co, see Yano [9; p. 67]. Let co be a (p, q)-form on a Kahler manifold M. Let 1

co =

v,

p ! q ! Lai w ... ap Pi ••• /14 d Zœl A

• • • A d ZIP A

d el A - •

A

d 21341

be a local expression for co. Then

q

1 d" a) = p! (q+1)!

E (vii c°44 Pi ••• Jig

. d z A A d2P A dgl A

— A

Ei VPk (DA A

— k

deg,

where A stands for oc1 ... al, and d z A for dza 1 A

1

... 4-1 Pok + 1... ff.)

••• A

deg. We have

VE0002... Ai d zA A d 92 A

••• A

d2Pq.

Hence, if co is a (0, 1)-form and

co = E wa d2Œ, then d" co =AE(Va coo Vii coa ) d? A dzs, —

6" d" co= — E v. (vet coo — v„ wŒ) d2 0

5"w= — E v. coa, d" eV co=

— E vi, sic, wa ay.

Hence, LI" co = (5" d" + d" S") co

= E ( — VŒ Va (Op ± Vet Vo (pa — Vo yi of) d-ifl =(—Va Vacopcig-FR Œpcoad20). It is known that, for a Kahler manifold,

A = 2 z1" . It is therefore possible to derive the fomula above for 4"w from the formula for d co.

4. A Remark on d' d"-Cohomology

159

4. A Remark on d' d" -Cohomology We prove the following well known Theorem 1. Let 0 and 0' be closed real (p, p)-forms on a compact Kahler manfiold M. Then 0 0' ( cohomologous to each other) if and only if there exists a real (p —1, p —1)-form 9 on M such that 0 — 0' = i d' d" (p .

Proof If 0 0' =i d' d" 9, then 0 — 0' = d (i d" (p) and hence 0 — 0'. (This implication is purely local and is valid for any complex manifold.) To prove the converse, let ri = 0— 0' and assume ri —0 so that ti =da, where a is a real (2p 1)-form. Let a =i3- + f3, where f3 is a (p — 1, p)-form and /1 is its complex conjugate. Then

where d' )4" is of bidegree (p +1, p-1), (d" + d' 13) of bidegree (p, p), and d" fi of bidegree (p —1, p + 1). Hence, t1=doc=d"

13,

d' )6=0,

d" )6=0.

We may write /3 =1-113 +d"y, where y is a (p —1, p — 1)-form. Then 4=11fl+d).

)

Hence, n=d"i3- ±d'13=d"d'7+d'd"y=d'd"(y—TI) i di d" 9,

where (p = — i(fi — /3-) is a real (p —1, p — 1)-form.

Bibliography

Accola, R. D. M. [1] Automorphisms of Riemann surfaces. J. Analyse Math. 18, 1-5 (1967) (MR 35 *4398). [2] On the number of automorphisms of a closed Riemann surfaces. Trans. Amer. Math. Soc. 131, 398-408 (1968) (MR 36 *5333). Aeppli, A. [1] Some differential geometric remarks on complex homogeneous manifolds. Arch. Math. (Basel) 16, 60-68 (1965) (MR 32 *464). Ambrose, W., Singer, I. M. [1] On homogeneous Riemannian manifolds. Duke Math. J. 25, 647-669 (1958) (MR 21 *1628).

Andreotti, A. [1] Sopra le superficie algebriche che posseggono tiansformazi.oni birazionali in se, Univ. Roma 1st. Naz. Alta Mat. Rend. Mat e Appl. 9, 255-279 (1950) (MR 14,20). Andreotti, A., Salmon, P. [1] Anelli con unica decomposabilità in fattori primi cd un problema di intersezione complete. Monatsh. Math. 61, 97-142 (1957) (MR 21 # 3415). Apte, M., Lichnerowicz, A. [1] Sur les transformations affines d'une variété presque hermitienne compacte. C. R. Acad. Sci. Paris Sér. A-B 242,337-339 (1956) (MR 17,787). Arens, R. [1] Topologies for hemeomorphism groups. Ann. of Math. 68, 593-610(1946) (MR 8, 479). [2] Normal forms for a Pfaffian. Pacific J. Math. 14,1-8 (1964) (MR 30 *2424).

Atiyah, M., Bott, R. [1] A Lefschetz fixed point formula for elliptic complexes. I. Ann. of Math. 86, 374-407 (1967) (MR 35 # 3701). IL Ann. of Math. 88, 451-491 (1968) (MR 38 *731). Atiyah, M., Hirzebruch, F. [1] Spin manifolds and group actions. Atiyah, M., Singer, L M. [1] The index of elliptic operators. III. Ann. of Math. 87, 536-604 (1968) (MR 38 *5245). Aubin, T. [1] Métriques riemanniennes et courbure. J. Differential Geometry 4,383-424 (1970).

Bibliography

161

Ba, B. [1] Structures presque complexes, structures conformes et dérivations. Cahiers Topologie Géom. Différentielle 8, 1-74 (1966) (MR 37 *5809). Barbance, C. [1] Transformations conformes d'une variété riemannienne compacte. C. R. Acad. Sci. Paris Sér. A-B 266, A149-152 (1968) (MR 38 *5137). Barros, C. de [I] Sur la géométrie différentielle des formes différentielles extérieures quadratiques. Atti Convergno Internaz. Geom. Diff. (Bologna), 1967,1-26. Baum, P. F., Bott, R. [I] On the zeroes of meromorphic vector fields. Essays on topology and related topics. Mémoires dédiés à G. de Rham, 1970,29-47 (MR 41 # 6248). Baum, P. F., Cheeger, J. [I] Infinitesimal isometries and Pontrjagin numbers. Topology 8, 173-193 (1969) (MR 38 # 6627). Berger, M. . Trois remarques sur les variétés riemanniennes à courbure positive. C. R. Acad. [1] Sci. Paris Sér. A-B 263, 76-78 (1966) (MR 33 *7966). [2] Sur les variétés d'Einstein. Comptes Rendus de la III` Réunion du Groupement des Mathématiciens d'Expression Latine, Namur (1965), 33-55 (MR 38 *6502). Bergmann, S. [I] Ober die Kernfunktion eines Bereiches, ihr Verhalten am Rande. J. Reine Angew. Math. 169, 1-42 (1933) and 172, 89-128 (1935). Bernard, D. [1] Sur la géométrie différentielle des G-structures. Ann. Inst. Fourier (Grenoble) 10, 151-270 (1960) (MR 23 *A4094). Bishop, R. L., Crittenden, R. J. [1] Geometry of manifolds. New York: Academic Press 1964 (MR 29 *6401). Bishop, R. L., O'Neill, B. [1] Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1-49(1969) (MR 40 #489). Blanchard, A. [I] Sur les variétés analytiques complexes. Ann. Sci. École Norm. Sup. 63, 157-202 (1958) (MR 19,316). Bochner, S. [1] Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52, 776-797 (1946) (MR 8, 230). [2] On compact complex manifolds. J. Indian Math. Soc. 11, 1-21 (1947) (MR 9,423). [3] Tensor fields with finite bases. Ann. of Math. 53, 400-411 (1951) (MR 12,750). Bochner, S., Martin, W. T. [I] Several complex variables. Princeton Univ. Press 1948 (MR 10,366). Bochner, S., Montgomery, D. [1] Locally compact groups of differentiable transformations. Ann. of Math. 47,639653 (1946) (MR 8,253). [2] Groups on analytic manifolds. Ann. of Math. 48,659-669 (1947) (MR 9,174). [3] Groups of differentiable and real or complex analytic transformations. Ann. of Math. 46,685-694 (1945) (MR 7,241).

162

Bibliography

Boothby, W. M. • [1] Homogeneous complex contact manifolds. Proc. Symp. Pure Math. vol. III, 144154, Amer. Math. Soc. 1961 (MR 23 #A2173). [2] A note on homogeneous complex contact manifolds. Proc. Amer. Math. Soc. 13, 276-280(1962) (MR 25 *582). [3] Transitivity of automorphisms of certain geometric structures. Trans. Amer. Math. Soc. 137, 93-100 (1969) (MR 38 * 5254). Boothby, W. M., Kobayashi, S., Wang, H. C. [1] A note on mappings and automorphisms of almost complex manifolds. Ann. of Math. 77, 329-334 (1963) (MR 26 *4284). Boothby, W. M., Wang, H. C. [1] On contact manifolds. Ann. of Math. 68, 721-734 (1958) (MR 22 43015). Borel, A. [1] Topology of Lie groups and characteristic classes. Bull. Amer. Math. Soc 61, 397432 (1955) (MR 17, 282). [2] Some remarks about Lie groups transitive on spheres and tori. Bull. Amer. Math. Soc. 55, 580-587 (1949) (MR 10, 680). [3] Seminar on transformation groups. Ann. of Math., Studies No. 46, (1960) (MR 22 *7129). [4] Groupes linéaires algébriques. Ann. of Math. 64, 20-82 (1956) (MR 19 #1195). [5] Linear algebraic groups. Benjamin 1969 (MR 40 *4273).

Borel, A., Narasimhan, R. [1] Uniqueness conditions for certain holomorphic mappings. Invent. Math. 2, 247255 (1967) (MR 35 *414). Borel, A., Remmert, R. [1] Ober kompakte homogene Kahlersche Mannigfaltigkeiten. Math. Ann. 145, 429439 (1961/62) (MR 26 #3088). Bott, R. [1] Vector fields and characteristic numbers. Michigan Math. J. 14, 231-244 (1967) (MR 35 *2297). [2] A residue formula for holomorphic vector fields. J. Differential Geometry 1, 311330 (1967) (MR 38 *730). [3] Non-degenerate critical manifolds. Ann. of Math. 60, 248-261 (1954) (MR 16, 276). Brickell, F. [1] Differentiable manifolds with an area measure. Canad. J. Math. 19, 540-549(1967) (MR 35 *2253). Busemann, H. [1] Similarities and differentiability. nhoku Math. J. 9, 56-67 (1957) (MR 20 *2772). [2] Spaces with finite groups of motions. J. Math. Pures Appl. 37, 365-373 (1958) (MR 21 *913). [3] The geometry of geodesics. Academic Press 1955 (MR 17, 779). Buttin, C. [1] Les dérivations des champs de tenseurs et l'invariant différentiel de Schouten. C. R. Acad. Sci. Paris Sér. A-B 269, A 87-89 (1969) (MR 40 #1913). [2] ttud2 d'un cas d'isomorphisme d'une algèbre de Lie filtrée avec son algèbre graduée associée. C. R. Acad. Sci. Paris Sér. A-B 264, A496-498 (1967) (MR 39 it 2826). Calabi, E. [1] On Kahler manifolds with vanishing canonical class. Algebraic geometry & topology. Symposium in honor of S. Lefschetz, 78-89, Princeton, 1957 (MR 19, 62).

Bibliography

163

Carathéodory, C. [1] Über die Abbildungen, die dutch Systeme von analytischen Funktionen von mehreren Verânderlichen erzeugt werden. Math. Z. 34, 758-792 (1932). Cartan, E. [1] Sur les variétés à connexion affine et la théorie de la relativité généralisée. Ann. Sci. École Norm. Sup. 40, 325-412 (1923); 41, 1-25 (1924), and 42, 17-88 (1925). [2] Les espaces à connexion conforme. Ann. Soc. Pa Math. 2, 171-221 (1923). [3] Sur un théorème fondamental de M. H. Weyl. J. Math. Pures Appl. 2, 167-192(1923). [4] Sur les variétés à connexion projective. Bull. Soc. Math. France 52, 205-241 (1924). [5] Sur la structure des groupes infinis de transformations. Ann. Sci. École Norm. Sup. 21, 153-206(1904), and 22, 219-308 (1905). [6] Les sous-groupes des groupes continus de transformations. Ann. Sci. École Norm. Sup. 25, 57-194 (1908). [7] Les groupes de transformations continus, infinis, simples. Ann. Sci. École Norm. Sup. 26, 93-161 (1909). [8] Leçons sur la Géométrie des Espaces de Riemann. Paris: Gauthier-Villars 1928; 2nd ed. 1946. [9] Leçons sur les invariants intégraux. Paris: Hermann 1922. Cartan, H. [1] Sur les groupes de transformations analytiques. Actualités Sci. Indust. No. 198. Paris: Hermann 1935. [2] Sur les fonctions de plusieurs variables complexes. L'itération des transformations intérieures d'un domaine borné, 'Math. 1 35, 760-773 (1932). Chern, S.S. [1] Pseudo-groupes continus infinis. Colloque de Géométrie Diff. Strasbourg (1953), 119-136 (MR 16, 112). [2] The geometry of G-structures. Bull. Amer. Math. Soc. 72, 167-219 (1966) (MR 33 *661). Chevalier, H.

[1] Noyau sur une variété analytique. C. R. Acad. Sci. Paris Set.. A-B 262, A 948-950 (1966) (MR 34 *3591). Chevalley, C. [1] Theory of Lie groups. Princeton Univ. Press 1946 (MR 7, 412). Chu, H., Kobayashi, S. [1] The automorphism group of a geometric structure. Trans. Amer. Math. Soc. 113, 141-150 (1963) (MR 29 # 1596). Conner, P. E. [1] On the action of the circle group. Michigan Math. J. 4, 241-247 (1957) (MR 20 *3230). Couty, R.

[1] Sur les transformations de variétés riemanniennes et kkhlériennes. Ann. Inst. Fourier (Grenoble) 9, 147-248 (1959) (MR 22 5 12488). [2] Transformations projective sur un espace d'Einstein complet C. R. Acad. Sci. Paris Sér. A-B 252, 109-1097 (1961) (MR 28 5569). [3] Formes projectives fermées sur un espace d'Einstein, C. R. Acad. Sci. Paris Sér. A-B 270, A 1249-1251 (1970) (MR 42 51008). Dantzig, D. van, Waerden, B. L. van der [1] Ober metrisch homogene RAume. Abh. Math. Sem. Univ. Hamburg 6, 374-376 (1928).

164

Bibliography

Dinghas, A. [1] Verzerrungsatze bei holomorphen Abbildungen von Hauptbereichen automorpher Gruppen mehrerer komplexer Verânderlicher in einer KAhler-Mannigfaltigkeit. S.-B. Heidelberger Akad. Wiss. Math.-Natur. KI. 1968, 1-21 (MR 38 *6109). Douglis, A., Nirenberg, L. [1] Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8, 503-583 (1955) (MR 17, 743). Ebin, D. G., Marsden, 1E. [1] Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102-163 (1970) (MR 42 # 6865). Eckmann, B. [1] Structures complexes et transformations infinitesimales. Convergno Intern. Geometria Differenziale, Italia, 1953, 176-184 (MR 16, 518). Fells, J., Jr. [1] A setting for global analysis. Bull. Amer. Math. Soc. 72, 751-807 (1966) (MR 34 *3590). Fells, J., Jr., Sampson, H. [1] Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86, 109-160 (1964) (MR 29 *1603). [2] Énergie et déformations en géométrie différentielle. Ann. Inst. Fourier (Grenoble) 14, 61-69 (1964) (MR 30 #2529). Egorov, L P. [1] On the order of the group of motions of spaces with affine connection. Dokl. Akad. Nauk SSSR 57, 867-870 (1947) (MR 9, 468). [2] On collineations in spaces with projective connection. Dokl. Akad. Nauk SSSR 61, 605-608 (1948) (MR 10, 211). [3] On the groups of motions of spaces with asymmetric affine connection. Dokl. Akad. Nauk SSSR 64, 621-624 (1949) (MR 10, 739). [4] On a strengthening of Fubini's theorem on the order of the group of motions of a Riemannian space. Dokl. Akad. Nauk SSSR 66, 793-796 (1949) (MR 11, 211). [5] On groups of motions of spaces with general asymmetrical affine connection. Dokl. Akad. Nauk SSSR 73, 265-267 (1950) (MR 12, 636). [6] Collineations of projectively connected spaces. Dokl. Akad. Nauk SSSR 80, 709712 (1951) (MR 14, 208). [7] A tensor characterization of 21„ of nonzero curvature with maximum mobility. Dokl. Akad. Nauk SSSR 84, 209-212 (1952) (MR 14, 318). [8] Maximally mobile L„ with a semi-symmetric connection. Dokl. Akad. Nauk SSSR 84, 433-435 (1952) (MR 14, 318). [9] Motions in spaces with affine connection. Dokl. Akad. Nauk SSSR 87, 693-696 (1952) (MR 16, 171). [10] Maximally mobile Riemannian spaces V4 of non constant curvature. Dokl. Akad. Nauk SSSR 103, 213-232 (1955) (MR 17, 405). [11] Motions in affinely connected spaces. Dokl. Akad. Nauk SSSR 87, 693-696 (1952) (MR 16, 170). [12] Maximally mobile Einstein spaces with non constant curvature. Dokl. Akad. Nauk SSSR 145,975-978 (1962) .(MR 25 *4472). [13] Equi-affine spaces of third lacunary. Dokl. Akad. Nauk SSSR 108, 1007-1010 (1956) (MR 18, 506). [14] Riemann spaces of second lacunary. Dokl. Akad. Nauk SSSR 111, 276-279(1956), (MR 19, 169).

Bibliography

165

[15] Riemannian spaces of the first three lacunary types in the geometric sense. Dokl. Akad. Nauk SSSR 150, 730-732 (1963) (MR 27 *5204 Ehresmann, C. [1] Les connexions infinitésimales dans un espace fibr6 différentiable. Colloque de Topologie, Bruxelles 1950, 29-55 (MR 13, 159). [2] Sur les variétés presque complexes. Proc. Intern. Congr. Math. Cambridge, Mass. 1950, vol. 2, 412-419 (MR 13, 547). [3] Introduction A la théorie des structures infinitésimales et des pseudo-groupes de Lie. Colloque de Géométrie Diff., Strasbourg, 1953, 97-110 (MR 16, 75). [4] Sur les pseudo-groupes de Lie de type fini. C. R. Acad. Sci. Paris Sér. A-B 246, 360-362 (1958) (MR 21 *97). Eisenhart, L. P. [1] Riemannian geometry. Princeton: Princeton Univ. Press 1949 (MR 11, 687). [2] Non-Riemannian Geometry. Amer. Soc. Co Iloq. Publ. 8, 1927. [3] Continuous groups of transformations. Princeton: Princeton Univ. Press. 1933. Floyd, E. E. [1] On periodic maps and the Euler characteristics of associated spaces. Trans. Amer. Math. Soc. 72,138-147 (1952) (MR 13, 673). Frankel, T. T. [1] Fixed points and torsions on Kahler manifolds. Ann. of Math. 70, 1-8 (1959) (MR 24 # A 1730). [2] Manifolds with positive curvature. Pacific J. Math. 11, 165-174 (1961) (MR 23 # A 600). [3] On theorem of Hurwitz and Bochner. J. Math. Mech. 15, 373-377 (1966) (MR 33 *675).

Freifield, C. [1] A conjecture concerning transitive subalgebras of Lie algebras. Bull. Amer. Math. Soc. 76, 331-333 (1970) (MR 40 *5794). Friilicher, A. [1] Zur Differentialgeometrie der komplexen Strukturen. Math. Ann. 129, 50-95(1955) (MR 16, 857). Fubini, G. [1] Sugli spazi che amettono un gruppo continuo di movimenti. Annali di Mat. 8, 39-81 (1903). Fujimoto, A. [1] On the structure tensor of G-structures. Mem. Coll. Sci. Univ. Kyoto Ser. A18, 157-169 (1960) (MR 22 # 8522). [2] On automorphisms of G-structures. J. Math. Kyoto Univ. 1, 1-20 (1961) (MR 25 #4459). [3] Theory of G-structures. A report in differential geometry [in Japanese], vol. 1, 1966. Fujimoto, H. [1] On the holomorphic automorphism groups of complex spaces. Nagoya Math. J. 33, 85-106 (1968) (MR 39 *3038). [2] On the automorphism group of a holomorphic fibre bundle over a complex space. Nagoya Math. J. 37, 91-106 (1970) (MR 41 *3809). [3] On holomorphic maps into a real Lie group of holomorphic transformations. Nagoya Math. J. 40, 139-146 (1970) (MR 42 *3310).

166

Bibliography

Godbillon, C. [1] Géométrie différentielle et mécanique analytique. Paris: Hermann 1969 (MR 39 43416). Goldberg, S. I. [1] Curvature and Homology. New York: Academic Press 1962 (MR 25 42537). Goldberg, S.I., Kobayashi, S. [1] The conformal transformation group of a compact Riemannian manifold. Amer. J. Math. 84, 170-174 (1962) (MR 25 43481). Gottschling, E. [1] Invarianten endlicher Gruppen und biholomorphe Abbildungen. Invent Math. 6, 315-326 (1969) (MR 39 #5834). [2] Reflections in bounded symmetric domains. Comm. Pure Appl. Math. 2 2 693-714 (1969) (MR 40 #4479). Grauert, H., Remmert, R. [1] Über kompakte homogene komplexe Mannigfaltigkeiten. Arch. Math. 13,498-507 (1962) (MR 26 43089).

Gray, J. W. [1] Some global properties of contact structure. Ann. of Math. 69, 421-450 (1959) (MR 22 #3016). Gromoll, D., Wolf, J. A. [1] Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature. Bull. Amer. Math. Soc. 77, 545-552 (1971).

Guillemin, V. [1] A Jordan-Holder decomposition for a certain class of infinite dimensional Lie algebras. J. Dif. Geometry 2, 313-345 (1966) (MR 41 48481). [2] Infinite dimensional primitive Lie algebra, J. Differential Geometry 4, 257-282 (1970) (MR 42 #3132). [3] Integrability problem for G-structures. Trans. Amer. Math. Soc. 116, 544-560 (1965) (MR 34 43475). Guillemin, V., Quillen, D., Sternberg, S. [1] The classification of the irreducible complex algebras of infinite type. J. Analyse Math. 18, 107-112 (1967) (MR 36 4221). Guillemin, V., Sternberg, S. [1] An algebraic model of transitive differential geometry. Bull. Amer. Math. Soc. 70, 16-47 (1964) (MR 30 # 533). [2] Deformation theory of pseudogroup structures. Mem. Amer. Math. Soc. no. 64 (1966) (MR 35 #2302). Gunning, R. C. [1] On Vitali's theorem for complex spaces with singularities. J. Math. Mech. 8 133141 (1959) (MR 21 46446). Gunning, R. C., Rossi, H. [1] Analytic functions of several complex variables. Englewood Cliffs, N.J.: PrenticeHall 1965 (MR 31 #4927). Hano, J.-I. [1] On affine transformations of a Riemannian manifold. Nagoya Math. J. 9, 99-109 (1955) (MR 17,891). [2] On compact complex coset spaces of reductive Lie groups. Proc. Amer. Math. Soc. 15, 159-163 (1964) (MR 28 # 1258).

Bibliography

167

Hano, J.-L, Kobayashi, S. [1] A fibering of a class of homogeneous complex manifolds. Trans. Amer. Math. Soc. 94, 233-243 (1960) (MR 22 #5990). Hano, L-L, Matsushima, Y. [1] Some studies of Kfihlerian homogeneous spaces. Nogoya Math. J. 11, 77-92 (1957) (MR 18,934). Morimoto, A. Hano, Note on the group of affine transformations of an aflinely connected manifold. [1] Nagoya Math. J. 8, 71-81 (1955) (MR 16,1053). Hartman, P. [1] On homotopic harmonic maps. Canad. J. Math. 19, 673-687 (1967) (MR 35 # 4856). Hatakeyama, Y. [1] Some notes on the group of automorphisms of contact and symplectic structures. Tôhoku Math. J. 18, 338-347 (1966) (MR 34 # 6679). Hawley, N. S. [1] A theorem on compact complex manifolds. Ann. of Math. 52, 637-641 (1950) (MR 12,603). Helgason, S. [1] Differential Geometry and Symmetric Spaces. New York: Academic Press 1962 (MR 26 #2986). Hermann, R. [1] Cartan connections and the equivalence problem for geometric structures. Contributions to Dif. Equations 3,199-248 (1964) (MR 29 #2741). Hirzebruch, F. [1] Topological Methods in Algebraic Geometry. Berlin-Heidelberg-New York: Springer 1966 (MR 34 # 2573). Hochschild, G. [1] The structure of Lie Groups. San Francisco: Holden Day 1965 (MR 34 #7696). Holmann, H. [1] Komplexe Raume mit komplexen Transformationsgruppen. Math. Ann. 150, 327360 (1963) (MR 27 # 776). Howard, A. [1] Holomorphic vector fields on algebraic manifolds. Amer. J. Math., to appear. Hsiang, W. Y. [1] The natural metric on SO(n)/S0(n - 2) is the most symmetric metric. Bull. Amer. Math. Soc. 73,55-58 (1967) (MR 35 #939). [2] On the bound of the dimensions of the isometry groups of all possible riemannian metrics on an exotic sphere. Ann. of Math. 85, 351-358 (1967) (MR 35 # 4935). [3] On the degree of symmetry and the structure of highly symmetric manifolds. Tamkang J. of Math., Tamkang College of Arts & Sc., Taipei, 2, 1-22 (1971). Hsiung, C. C [1] Vector fields and infinitesimal transformations on Riemannian manifolds with boundary. Bull. Soc. Math. France 92, 411-434 (1964) (MR 31 # 2693).

Bibliography

168

Huang, W. H. [1] Lefschetz numbers for Riemannian manifolds. To appear. Hurwitz, A. [1] Ober algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41, 403-442 (1893). [2] Ober diejenigen algebraischen Gebilde, welche eindeutige Transformationen in sich zu lassen. Math. Ann. 32, 290-308 (1887). Illusie, L. [1] Nombres de Chem et groupes finis. Topology 7, 255-269 (1968) (MR 37 #4822). Ishida, M. [1] On groups of automorphisms of algebraic varieties. J. Math. Soc. Japan 14,276-283 (1962) (MR 26 # 1315). Ishihara, S. [1] Homogeneous Riemannian spaces of four dimension. J. Math. Soc. Japan 7, 345370 (1955) (MR 18, 599). [2] Groups of projective transformations on a projectively connected manifold. Japan. J. Math. 25, 37-80 (1955) (MR 18, 599). [3] Groups of projective transformations and groups of conformal transformations. J. Math. Soc. Japan 9, 195-227 (1957) (MR 20 *311). Ishihara, S., Obata, M. [1] A ff ine transformations in a Riemannian manifold. Tôhoku Math. J. 7, 146-150 (1955) (MR 17, 1128). .

Jfinich, K. [1] Differentierbare G-Mannigfaltigkeiten. Lectures notes in Math. No. 59. BerlinHeidelberg-New York: Springer 1968 (MR 37 *4835).

Jensen, G.R. [1] Homogeneous Einstein spaces of dimension four. J. Differential Geometry 3, 309350 (1969) (MR 41 *6100). [2] The scalar curvature of left-invariant Riemannian metrics. Indiana Univ. Math. J. 20, 1125-1144 (1971). Kac, V. G. [1] Simple graded Lie algebras of finite growth. Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1323-1367 (1968) (MR 41 # 4590). Kaneyuki, S. [1] On the automorphism groups of homogeneous bounded domains. J. Fac. Sci. Univ. Tokyo, Sect. 114, 89-130 (1967) (MR 37 *3056).

Kantor, I. L. [1] Infinite dimensional simple graded Lie algebras. Dokl. Akad. Nauk SSSR 179, 534-537 (1968) (MR 37 *1418). Soviet Math. Dokl. 9, 409-412 (1968). [2] Transitive differential groups and invariant connections in homogeneous spaces. Trudy Sem. Vektor. Tenzor. Anal. 13, 310-398 (1966) (MR 36 #1314).

Kantor, L. L., Sirota, A. I., Solodovnikov, A. S. [1] A class of symmetric spaces with an extensive group of motions and a generalization of the Poincaré model. Dokl. Akad. Nauk SSSR 173, 511-514 (1967). Soviet Math. Dokl. 8, 423-426 (1967) (MR 35 #3607).

Bibliography

169

Kaup, W. [1] Reelle Transformationsgruppen und invariante Metriken auf komplexen RAumen. Invent. Math. 3,43-70(1967) (MR 35 # 6865). [2] Holomorphe Abbildungen in hyperbolische }Lamm Centro Internaz. Mat. Estivo, 1967, pp. 111-123 (MR 39 *478). [3] Holomorphic mappings of complex spaces. Symp. Math., vol. II (INDAM, Roma, 1968), 333-340. Academic Press 1969 (MR 40 *5906). [4] Hyperbolische komplexe Rdume. Ann. Inst. Fourier (Grenoble) 18, 303-330(1968) (MR 39 #7138). Kaup, W., Matsushima, Y., Ochiai, T. [1] On the automorphisms and equivalences of generalized Siegel domains. Amer. J. Math. 92, 475-498 (1970) (MR 42 #2029). Kerner, H. • [1] Ober die Automorphismengruppen kompakter komplexer Mum. Arch. Math. 11, 282-288 (1960) (MR 22 *8533). Klingenberg, W. [1] Eine Kennzeichnung der Riemannschen sowie der Hermiteschen Mannigfaltigkeiten. Math. Z. 70, 300-309 (1959) (MR 21 # 3891). Knebelman, M. S. [1] Collineations of projectively related affine connections. Ann. of Math. 29, 389-394 (1928). [2] Collineations and motions in generalized spaces. Amer. J. Math. 51, 527-564 (1929). [3] On groups of motions in related spaces. Amer. J. Math. 52, 280-282 (1930). Kobayashi, E. T. [1] A remark on the Nijenhuis tensor. Pacific J. Math. 12, 963-977 and 1467 (1962) (MR 27 *678). [2] A remark on the existence of a G-structure. Proc. Amer. Math. Soc. 16, 1329-1331 (1965) (MR 35 *2235). Kobayashi, S. [1] Le groupe de transformations qui laissent invariant un parallélisme. Colloque de Topologie. Strasbourg: Ehresmann 1954 (MR 19,576). [2] Groupes de transformations qui laissent invariante une connexion infinitésimale, C. R. Acad. Sci. Paris Sér. A-B 238, 644-645 (1954) (MR 15,742). [3] A theorem on the affine transformation group of a Riemannian manifold. Nagoya Math. J. 9, 39-41 (1955) (MR 17,892). [4] Theory of connections. Ann. Mat. Pura Appl. 43, 119-194 (1957) (MR 20 * 2760). [5] Fixed points of isometries. Nagoya Math. J. 13, 63-68 (1958) (MR 21 *2276). [6] Geometry of bounded domains. Trans. Amer. Math. Soc. 92, 267-290 (1959) (MR 22 *3017). [7] On the automorphism group of a certain class of algebraic manifolds. Tôhoku Math. J. 11, 184-190 (1959) (MR 22 *3014). [8] Frame bundles of higher order contact. Proc. Symp. Pure Math. vol. 3, Amer. Math. Soc. 1961,186-193 (MR 23 *A4104). [9] Invariant distances on complex manifolds and holomorphic mappings. J. Math. Soc. Japan 19, 460-480 (1967) (MR 38 #736). [10] Hyperbolic manifolds and holomorphic mappings. New York: Marcel Dekker 1970.

170

Bibliography

Kobayashi, S., Nagano, T. [1] On projective connections. J. Math. Mech. 13, 215-236 (1964) (MR 28 # 2504 [2] On a fundamental theorem of Weyl-Cartan on G-structures. J. Math. Soc. Japan 17, 84-101 (1965) (MR 33 *663). [3] On filtered Lie algebras and geometric structures. I. J. Math. Mech. 13, 875-908 (1964) (MR 29 *5961); II. 14, 513-522 (1965) (MR 32 *2512); III. 14, 679-706 (1965) (MR 32 #5803); IV. 15, 163-175 (1966) (MR 33 *4189); V. 15, 315-328 (1966) (MR 33 *4189). [4] A report on filtered Lie algebras. Proc. US-Japan Seminar in Differential Geometry, Kyoto, 1965, 63-70 (MR 35 *6722). [5] Riemannian manifolds with abundant isometries. Dill Geometry in honor of K. Yano, Kinokuniya, Tokyo, 1972,195-220. Kobayashi, S., Nomizu, K. [1] Foundations of differential geometry. New York: John Wiley & Sons, vol. 1, 1963 (MR 27 #2945); vol. 2, 1969 (MR 38 # 6501). [2] On automorphisms of a Kahlerian structure. Nagoya Math. J. 11, 115-124 (1957) (MR 20 #4004). Kobayashi, S., Ochiai, T. [1] G-structures of order two and transgression operators. J. Differential Geometry 6, 213-230 (1971). [2] Mappings into compact complex manifolds with negative first Chun class. J. Math. Soc. Japan 23, 137-148 (1971). Kodaira, K. [1] On differential geometric method in the theory of analytic stacks. Proc. Nat. Acad. Sci. U.S.A. 39, 1263-1273 (1953) (MR 16, 618). [2] On Uhler varieties of restricted type. Ann. of Math. 60, 28-48 (1954)(MR 16, 252). [3] On deformations of some complex pseudogroup structures. Ann. of Math. 71, 224302 (1960) (MR 22 *599). Kosmann, Y. [1] Dérvées de Lie des spineurs. C. R. Acad. Sci. Paris, Sér. A-B 262, A 289-292 (1966) (MR 34 #723); Applications 262, A 394-397 (1966) (MR 34 #724). Kostant, B. [1] Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold. Trans. Amer. Math. Soc. 80, 528-542 (1955) (MR 18, 930). [2] On holonomy and homogeneous spaces. Nagoya Math. J. 12,31-54 (1957) (MR 21 *6003). [3] A characterization of invariant affine connections. Nagoya Math. J. 16, 35-50 (1960) (MR 22 # 1863). Koszul, IL. [1] Sur la forme hermitienne canonique des espaces homogènes complexes. Canad. J. Math. 7, 562-576(1955) (MR 17, 1109). [2] Domaines bornés homogènes et orbites de groupes de transformations affines. Bull. Soc. Math. France 89, 515-533 (1961) (MR 26 *3090). [3] Lectures on groups of transformations. Lectures on Math. & Physics, No. 20, Tata Inst. Bombay, 1965 (MR 36 *1571).

Ku, H.T., Mann, L. N., Sicks, J.L., Su, J. C. [1] Degree of symmetry of â product manifold. Trans. Amer. Math. Soc. 146, 133-149 (1969) (MR 40 # 3579).

Bibliography

171

Kuiper, N. H. [1] On conformally flat spaces in the large. Ann. of Math. 50,916-924 (1949) (MR 11, 133). [2] On compact conformally Euclidean spaces of dimension >2. Ann. of Math. 52 478-490 (1950) (MR 12,283). [3] Groups of motions of order 4n(n-1)+ 1 in Riemannian n-spaces. Indag. Math. 18, 313-318 (1955) (MR 18,232). Kuiper, N. H., Yano, K. [1] Two algebraic theorems with applications. Indag. Math. 18, 319-328 (1956) (MR 18,5). Kulkarni, R. S. . [1] Curvature structures and conformal transformations. J. Differential Geometry 4, 425-452 (1970). [2] On a theorem of F. Schur, J. Differential Geometry 4, 453-456 (1970). Kuranishi, M. [1] On the local theory of continuous infinite pseudogroups. I. Nagoya Math. J. 15, 225-260 (1959) (MR 22 *6866); II. 19, 55-91 (1961) (MR 26 *263). Kurita, M. [1] A note on conformal mappings of certain Riemannian manifolds. Nagoya Math. J. 21, 111-114 (1962) (MR 26 *718). [2] On the isometry of a homogeneous Riemann space. Tensor 3, 91-100 (1954) (MR 16,72). Ledger, A.J., Obata, M. [1] Compact riemannian manifolds with essential groups of conformorphisms. Trans. Amer. Math. Soc. 150, 645-651 (1970) (MR 41 *7576).

Lee, E. H. [1] On even-diemsnional skew symmetric spaces and their groups of transformations. Amer. J. Math. 67,321-328 (1945). Lefebvre, J. [1] Propriétés de groupe des transformations conformes et du groupe des automorphismes d'une variété localement conformement symplectique. C. R. Acad. Sci. Paris Sér. A-B 268, A 717-719 (1969) (MR 39 *6199). [2] Propriétés des algèbres d'automorphismes de certaines structures presque symplectiques. C. R. Acad. Sci. Paris Sér. A-B 266, A 354-356 (1968) (MR 37 *6883). [3] Transformations conformes et automorphismes de certaines structures presque symplectiques. C. R. Acad. Sci. Paris Sér. A-B 262, A 752-754 (1966) (MR 34 *1954). Lehmann-Lejeune, J. [1] Intégrabilité des G-structures définies par une 1-forme 0-déformable i valeurs dans le fibre tangent. Ann. Inst. Fourier (Grenoble) 16, 329-387 (1966) (MR 35 *3586). Lehner, J., Newman, M. [1] On Riemann surfaces with maximal automorphism groups. Glasgow Math. J. 8, 102-112 (1967) (MR 36 *3976).

Lelong, P. [1] Intégration sur un ensemble analytique complexe. Bull. Soc Math. France 85, 239-262 (1957) (MR 20 *2465).

172

Bibliography

Lelong-Ferrand, J. [1] Quelques propriétés des groupes de transformations infinitésimales d'une variété riemannienne. Bull. Soc. Math. Belg. 8, 15-30(1956) (MR 19,168). [2] Sur les groupes à un parametre de transformations des variétés différentiables. J. Math. Pures Appl. 37, 269-278 (1958) (MR 20 *4874). Sur l'application des méthodes de Hilbert à l'étude des transformations infini[3] tésimales d'une variété différentiable. Bull. Soc. Math. Belg. 9, 59-73 (1957) (MR 21 *2274). [4] Transformations conformes et quasiconformes des variétés riemanniennes; applications à la démonstration d'une conjecture de A. Lichnerowicz. C. R. Acad. Sci. Paris Sér. A-B 269, A 583-586 (1969) (MR 40 *7989). Leslie, J. A. [1] On a differential structure for the group of diffeomorphisms. Topology 6,263-271 (1967) (MR 35 *1041). [2] Two classes of classical subgroups of Diff(M). J. Differential Geometry 5, 427-435 (1971). Lewittes, J. [1] Automorphisms of compact Riemann surfaces. Amer. J. Math. 85, 734-752 (1963) (MR 28 *4102). Libermann, P. [1] Sur le problème d'équivalence de certaines structures infinitésimales. Ann. Mat. Pura Appl. 36, 27-120 (1954) (MR 16,520). [2] Sur les automorphismes infinitésimaux des structures symplectiques et des structures contactes, Colloque Géométrie Duff. Globale, Bruxelles, 1958,37-59 (MR 22 *9919). [3] Pseudogroupes infinitésimaux attachés aux pseudogroupes de Lie. Bull. Soc. Math. France 87, 409-425 (1959) (MR 23 # A 607). Lichnerowicz, A. [1] Géométrie des groupes de transformations. Paris: Dunod 1958 (MR 23 *A 1329). [2] Isométrie et transformations analytiques d'une variété kAhlérienne compacte. Bull. Soc. Math. France 87,427-437 (1959) (MR 22 # 5012). [3] Sur les transformations analytiques d'une variété kNhlérienne compacte. Colloque Geom. Diff. Global, Bruxelles, 1958,11-26 (MR 22 #7150). [4] Théorèmes de réductivité sur des algèbres d'automorphismes. Rend. Mat. e Appl. 22, 197-244 (1963) (MR 28 #544). [5] Variétés complexes et tenseur de Bergmann. Ann. Inst. Fourier (Grenoble) 15, 345407 (1965) (MR 33 *744). [6] Variétés kNhlériennes et première classe de Chern. J. Differential Geometry 1, 195-224 (1967) (MR 37 #2150). [7] Transformations des variétés à connexion linéaire et des variétés riemannienne. Enseignement Math. 8, 1-15 (1962) (MR 26 *717). [8] Zéros des vecteurs holomorphes sur une variété kAhlérienne a première classe de Chem non-négative. Diff. Geometry in honor of K. Yano, Kinokuniya, Tokyo, 1972,253-266. [9] Variétés kailériennes a première classe de Chem non-négative et variétés riemanniennes à courbure de Ricci généralisée non-négative. J. Differential Geometry 6, 47-94 (1971). Loos, O. [1] Symmetric spaces, 2 volumes. New York: Benjamin 1969 (MR 39 # 365).

Bibliography

173

Lusztig, G. [1] A property of certain non-degenerate holomorphic vector fields. An. Univ. Timisoara, Ser. $ti. Mat.-Fiz. 7, 73-77 (1969) (MR 42 #2032). Macbeath, A. M. [1] On a theorem of Hurwitz. Proc. Glasgow Math. Assoc. 5, 90-96 (1961) (MR 26 *4244).

Mann, L. N. [1] Gaps in the dimensions of transformation groups. Illinois J. Math. 10, 532-546 (1966) (MR 34 # 282). Manturov, O. V. [1] Homogeneous Riemannian spaces with an irreducible rotation group. Trudy Sem. Vektor. Tenzor. Anal. 13, 68-145 (1966) (MR 35 # 926). Matsumura, H. [1] On algebraic groups of birational transformations. Accad. Naz. dei Lincei 34, 151-155 (1963) (MR 28 #3041). Matsumura, H., Monsky, P. [1] On the automorphisms of hypersurfaces. J. Math. Kyoto Univ. 3-3, 347-361 (1964) - (MR 29 #5819). Matsushima, Y. [1] Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kAhlérienne. Nagoya Math. J. 11, 145-150 (1957) (MR 20 *995). [2] Sur les espaces homogènes kfihlériens d'un groupe de Lie réductif. Nagoya Math. J. 11, 53-60(1957) (MR 19, 315). [3] Sur certaines variétés homogènes complexes. Nagoya Math. J. 18, 1-12 (1961) (MR 25 #2147). [4] Holomorphic vector fi elds and the first Chern class of a Hodge manifold. J. Differential Geometry 3, 477-480 (1969) (MR 42 #843). [5] Holomorphic vector fields on compact Uhler manifolds. Cog. Board Math. Sci. Regional Cog. Ser. in Math. No. 7, 1971, Amer. Math. Soc. [6] On Hodge manifolds with zero first Chem class. J. Differential Geometry 3, 477480 (1969). Milnor, J. [1] Lectures on Morse theory. Annals Math. Studies No. 51. Princeton Univ. Press 1963 (MR 29 # 634).

Montgomery, D., Samelson, H. [1] Transformation groups of spheres. Ann. of Math. 44, 454-470 (1943) (MR 5, 60). Montgomery, D., Zippin, L. [1] Transformation groups. Interscience tracts #1. J. Wiley & Sons 1955 (MR 17, 383). Morimoto, A. [1] Sur le groupe d'automorphismes d'un espace fibré principal analytique complexe. Nagoya Math. J. 13, 157-178 (1958) (MR 20 # 2474). [2] Prolongations of G-structures to tangent bundles. Nagoya Math. J. 32, 67-108 (1968) (MR 37 # 6865). [3] Prolongations of G-structures to tangent bundles of higher order. Nagoya Math. J. 38, 153-179 (1970) (MR 41 *7568). [4] Prolongations of connections to tangential fibre bundles of higher order. Nagoya . Math. J. 40, 85-97 (1970).

174

Bibliography

Morimoto, A., Nagano, T. [1] On pseudo-conformal transformations of hypersurfaces. J. Math. Soc. Japan 15, 289-300 (1963) (MR 27 *5275). Morimoto, T., Tanaka, N. [1] The classification of the real primitive infinite Lie algebras. J. Math. Kyoto Univ. 10-2, 207-243 (1970). Moser, J. [1] On the volume element on a manifold. Trans. Amer. Math. Soc. 120, 286-294 (1965) (MR 32 *409).

Muta, Y. [1] On n-dimensional projectively flat spaces admitting a group of affine motions G,. of order r > n. Sci. Rep. Yokohama Nat. Univ. Sec. 1 4, 3-18 (1955) (MR 17,783). [2] On the curvature affinor of an affinely connected manifold A„, n 7 admitting a group of affine motions G, of order r > — 2 n. Tensor 5, 39-53 (1955) (MR 17,407). Myers, S. B., Steenrod, N. [1] The group of isometrics of a Riemannian manifold. Ann. of Math. 40, 400-416 (1939).

Nagano, T. [1] The conformal transformation on a space with parallel Ricci tensor. J. Math. Soc. Japan 11, 10-14 (1959) (MR 23 +A1330). [2] On conformal transformations in Riemannian spaces. J. Math. Soc. Japan 10, 79-93 (1958) (MR 22 1859). [3] Sur des hypersurfaces et quelques groupes d'isometries d'un éspace riemannien. Tahoku Math. J. 10, 242-252 (1958) (MR 21 *344). [4] Transformation groups with (n — 1)-dimensional orbits on noncompact manifolds. Nagoya Math. J. 14, 25-38 (1959) (MR 21 *3513). [5] On some compact transformation groups on spheres. Sci. Papers College Gen. Ed. Univ. Tokyo 9, 213-218 (1959) (MR 22 *4799). [6] Homogeneous sphere bundles and the isotropic Riemann manifolds. Nagoya Math. J. 15,29-55 (1959) (MR 21 *7522). [7] Transformation groups on compact symmetric spaces. Trans. Amer. Math. Soc. 118, 428-453 (1965) (MR 32 *419). [8] The projective transformation on a space with parallel Ricci tensor. Kadai Math. Sem. Rep. 11,131-138 (1959) (MR 22 216). [9] Isometrics on complex-product spaces. Tensor 9, 47-61 (1959) (MR 21 *6599), [10] Linear differential systems with singularities and application to transitive Lie algebras. J. Math. Soc. Japan 18, 398-404 (1966) (MR 33 *8005). [11] 1-forms with the exterior derivatives of maximal rank. J. Differential Geometry 2, 253-264(1968) (MR 39 *883). Nakano, S. [1] On complex analytic vector bundles. J. Math. Soc. Japan 7,1-12 (1955) (MR 17, 409).

Narasimhan, M. S., Simha, R. R. [1] Manifolds with ample canonical class. Invent. Math. 5, 120-128 (1968) (MR 38 45253). Naruki, I. [1] A note on the automorphism group of an almost complex structure of type (n, n'). Proc. Japan Acad. 44, 243-245 (1968) (MR 37 *3586).

Bibliography

175

Newlander, A., Nirenberg, L. [1] Complex analytic coordinates in almost complex manifolds. Ann. of Math. 65, 391-404 (1957) (MR 19,577). Nijen huis, A., Woolf; W. B. [1] Some integration problems in almost complex manifolds. Ann. of Math. 77, 424483 (1963) (MR 26 46992). Nomizu, K.

[1] On the group of affine transformations of an affinely connected manifold. Proc. Amer. Math. Soc. 4, 816-823 (1953) (MR 15,468). [2] Invariant affine connections on homogeneous spaces. Amer. J. Math. 76, 33-65 [3] [4] [5] [6] [7]

(1954) (MR 15,468). Lie groups and differential geometry. Publ. Math. Soc. Japan # 2,1956 (MR 18,821). On local and global existence of Killing vector fields. Ann. of Math. 72 105-120 (1960) (MR 22 #9938). Sur les algèbres de Lie de générateurs de Killing et l'homogénéité d'une variété riemannienne. Osaka Math. J. 14, 45-51 (1962) (MR 25 44467). Holonomy, Ricci tensor and Killing vector fi elds. Proc. Amer. Math. Soc. 12, 594597 (1961) (MR 24 4A 1099). Recent development in the theory of connections and holonomy groups. Advances in Math., fasc. 1,1-49 New York: Academic Press (MR 25 45473).

Nomizu, K., Yano, K. [1] On infinitesimal transformations preserving the curvature tensor field and its covariant differentials. Ann. Inst. Fourier (Grenoble) 14, 227-236 (1964) (MR 30 #4227). [2] Some results related to the equivalence problem in Riemannian geometry. Proc. US-Japan Seminar in Diff. Geometry, Kyoto, Japan, 1965,95-100 (MR 37 #2136); Math. Z. 97, 29-37 (1967) (MR 37 #2137). Obata, M. [1] Affine transformations in an almost complex manifold with a natural affine connection. J. Math. Soc. Japan 8, 345-362 (1956) (MR 18,822). [2] On the subgroups of the orthogonal groups. Trans. Amer. Math. Soc. 87, 347-358 (1958) (MR 20 #1711). [3] Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333-340 (1962) (MR 25 45479). [4] Conformal transformations of compact Rièmannian manifolds. Illinois J. Math. 6, 292-295 (1962) (MR 25 4 1507). [5] Riemannian manifolds admitting a solution of a certain system of differential equations. Proc. US-Japan Seminar in Diff. Geometry, Kyoto, Japan, 1965,101114 (MR 35 #7263). [6] Conformal transformations of Riemannian manifolds. J. Differential Geometry 4, 331-334 (1970) (MR 42 #2387). [7] Conformaily flat Riemannian manifolds admitting a one-parameter group of conformal transformations. J. Differential Geometry 4, 335-338 (1970) (MR 42 #2388). [8] The conjectures on conformal transformations of Riemannian manifolds. Bull. Amer. Math. Soc. 77, 265-270 (1971) (MR 42 #5286). J. Differential Geometry 6, 247-258 (1971). [9] Conformal changes of Riemannian metrics on a Euclidean sphere. Diff. Geometry in honor of K. Yano, Kinokuniya, Tokyo, 345-354 (1972). [10] On n-dimensional homogeneous spaces of Lie groups of dimension greater than n(n — 1)/2. J. Math. Soc. Japan 7, 371-388 (1955) (MR 18,599).

176

Bibliography

Obata, M., Ledger, Ai. [1] Transformations conformes des variétés riemanniennes compactes. C. R. Acad. Sci. Paris Sér. A-B 270, A459-461 (1970) (MR 40 *7990). Ochiai, T. [1] Classification of the finite nonlinear primitive Lie algebras. Trans. Amer. Math. Soc. 124, 313-322 (1966) (MR 34 *4320). [2] On the automorphism group of a G-structure. J. Math. Soc. Japan 18, 189-193 . (1966) (MR 33 4V 3224). [3] Geometry associated with semi-simple flat homogeneous spaces. Trans. Amer. Math. Soc. 152, 1-33 (1970). Oeljeklaus, E. [1] Ober fasthomogene kompakte Mannigfaltigkeiten. Schr. Math. Inst. Univ. Milnster (2) Heft 1 (1970) (MR 42 *2033). Ogiue, K. [1] Theory of conformal connections. Kifidai Math. Sent Rep. 19, 193-224 (1967) (MR 36 *812). [2] G-structures of higher order. KUdai Math. Sem. Rep. 19, 488-497 (1967) (MR 36 *814). Omori, H. [1] On the group of diffeomorphisms on a compact manifold. Proc. Symp. Pure Math. XV, Amer. Math. Soc., 1970,167-183 (MR 42 *6864). [2] Groups of diffeomorphisms and their subgroups, to appear. [3] On contact transformation groups. J. Math. Soc. Japan, to appear. Ozols, V. [1] Critical points of the displacement functions of an isometry. J. Differential Geometry 3,411-432 (1969) (MR 42 *1010). Palais, R. [1] A global formulation of the Lie theory of transformation groups. Mem. Amer. Math. Soc. *22,1957 (MR 22 *12162). [2] On the differentiability of isometries. Proc. Amer. Math. Soc. 8, 805-807 (1957) (MR 19,451). Peters, K. [1] Ober holomorphe und meromorphe Abbildungen gewisser kompakter komplexer Mannigfaltigkeiten. Arch. Math. (Basel) 15, 222-231 (1964) (MR 29 *4071). Pjatetzki-Shapiro, I. I. [1] Géométrie des domaines classiques et théorie des fonctions automorphes. Paris: Dunod 1966 (MR 33 *5949). Poincaré, H. [1] Sur un théorème de M. Fuchs. Acta Math. 7, 1-32 (1885). Preismann, A. [1] Quelques propriétés globales des espaces de Riemann. Comment. Math. Helv. 15, 175-216 (1942) (MR 6,20). Ra§evskii, P. K. [1] On the geometry of homogeneous spaces. Dokl. Akad. Nauk SSSR 80, 169-171 (1951) (MR 13,383); Trudy Sent Vektor. Tenzor. Anal. 9, 49-74 (1952) (MR 14, 795).

Bibliography

177

Renunert, R. [1] Holomorphe und meromorphe Abbildungen komplexer RNume. Math. Ann. 133, 328-370 (1957) (MR 19, 1193). Rodrigues, A. M. [1] The first and second fundamental theorems of Lie for Lie pseudo groups. Amer. J. Math. 84, 265-282 (1962) (MR 26 #5102). [2] On Cartan pseudo groups. Nagoya Math. J. 23, 1-4 (1963) (MR 30 #4862).

Rossi, H. [1] Vector fields on analytic spaces. Ann. of Math. 78, 455-467 (1963) (MR 29 # 277). Ruh, E.A. [1] On the automorphism groups of a G-structure. Comment. Math. Helv. 39, 189-264 (1964) (MR 31 #1631). Samelson, H. [1] Topology of Lie groups. Bull. Amer. Math. Soc. 58, 2-37 (1952) (MR 13 #533). Sampson, H. [1] A note on automorphic varieties. Proc. Nat. Acad. Sci. U.S.A. 38, 895-898 (1952) (MR 14, 633). Sasaki, S. [1] Almost contact manifolds. Math. Inst. Tôhoku Univ., Part I, Part II, 1967; Part III, 1968. Sawaki, S. [1] A generalization of Matsushima's theorem. Math. Ann. 146, 279-286 (1962) (MR 25 #538).

Schwarz, H. A. [1] Ober diejenigen algebraischen Gleichungen zwischen zwei verânderlichen Gralen, welche eine Schar rationaler eindeutig umkehrbarer Transformationen in sich selbst zulassen. Crelle's J. 87, 139-145 (1879). Shnider, S. [1] The classification of real primitive infinite Lie algebras. J. Differential Geometry 4, 81-89 (1970). Singer, I. M. [1] Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13, 685-697 (1960) (MR 24 # A 1100). Singer, L M., Sternberg, S. [1] On the infinite groups of Lie and Cartan. L Ann. Inst Fourier (Grenoble) 15, 1-114 (1965) (MR 36 #991). lebodzifiski, A. [1] Formes extérieurs et leurs applications. Warzawa, vol. 1, 1954 (MR 16, 1082); vol. 2, 1963 (MR 29 # 2738). Solodovnikov, [1] The group of projective transformations in a complete analytic Riemannian space. Dokl. Akad. Nauk SSSR 186, 1262-1265 (1969); Soviet Math. Dokl. 10, 750-753 (1969) (MR 40 #1936). Spencer, D. C. [1] Deformation of structures on manifolds defined by transitive, continuous pseudogroups. Part I. Ann. of Math. 76, 306-398 (1962); Part II, 399-445 (MR 27 # 6287).

178

Bibliography

Sternberg, S. [1] Lectures on differential geometry. Prentice-Hall 1964 (MR 23 *1797). Suyama, Y., Tsukamoto, Y. [1] Riemannian manifolds admitting a certain conformal transformation group. J. Differential Geometry 5, 415-426 (1971). Synge, J. L. [1] On the connectivity of spaces of positive curvature. Quart. J. Math. 7, 316-320 (1936). Takeuchi, M. [1] On the fundamental group and the group of isometrics of a symmetric space. J. Fac. Sci. Univ. Tokyo Sect. I10, 88-123 (1964) (MR 30 # 1217).

Takizawa, S. [1] On contact structures of real and complex manifolds. Tôhoku Math. J. 15, 227252 (1963) (MR 28 # 566). Tanaka, N. [1] Conformal connections and conformal transformations. Trans. Amer. Math. Soc. 92, 168-190 (1959) (MR 23 A 1331). [2] Projective connections and projective transformations. Nagoya Math. J. 11, 1-24 (1957) (MR 21 *3899). [3] On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables. J. Math. Soc. Japan 14, 397-429 (1962) (MR 26 *3086). [4] On the equivalence problems associated with a certain class of homogeneous spaces. J. Math. Soc. Japan 17, 103-139 (1965) (MR 32 # 6358). [5] Graded Lie algebras and geometric structures. Proc. US-Japan Seminar in Diff. Geometry, Kyoto, 1965, 147-150 (MR 36 # 5852). [6] On generalized graded Lie algebras and geometric structures. L J. Math. Soc. Japan 19, 215-254 (1967) (MR 36 *4470). [7] On differential systems, graded Lie algebras and pseudo-groups. J. Math. Kyoto Univ. 10, 1-82 (1970) (MR 42 # 1165). [8] On infinitesimal automorphisms of Siegel domains. J. Math. Soc. Japan 2 2 180212 (1970) (MR 42 # 7939).

Tanno, S. [1] Strongly curvature-preserving transformations of pseudo-Riemannian manifolds. Tôhoku Math. J. 19, 245-250 (1967) (MR 36 # 3284). [2] Transformations of pseudo-Riemannian manifolds. J. Math. Soc. Japan 21, 270281 (1969) (MR 39 # 2101). The automorphism groups of almost Hermitian manifolds. Trans. Amer. Math. [3] Soc. 137, 269-275 (1969) (MR 38 *5144). Tanno, S., Weber, W. C. [1] Closed conformal vector fields. J. Differential Geometry 3, 361-366 (1969) (MR 41 *6111). Tashiro, Y. [1] Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251-271 (1965) (MR 30 # 4229). [2] Conformal transformations in complete Riemannian manifolds. PubL Study Group of Geometry, vol. 3, Kyoto Univ. 1967 (MR 38 *639). [3] On projective transformations of Riemannian manifolds. J. Math. Soc. Japan 11, 196-204 (1959) (MR 23 # A 602).

Bibliography

179

Tashiro, Y., Miyashita, K. [1] Conformal transformations in complete product Riemannian manifolds. J. Math. Soc. Japan 19, 328-346 (1967) (MR 35 *3605). [2] On conformal diffeomorphisms of complete Riemannian manifolds with parallel Ricci tensor. J. Math. Soc. Japan 23, 1-10(1971). Teleman, C. [1] Sur les groupes maximums de mouvement des espaces de Riemann K. Acad. R. P. Romine Stud. Cerc. Mat. 5, 143-171 (1954) (MR 16, 624). [2] Sur les groupes des mouvements d'un espace de Riemann. J. Math. Soc. Japan 15, 134-158 (1963) (MR 27 *6219). Veisfeiler, B. Ju. [1] On filtered Lie algebras and their associated graded algebras. FunkcionaL AnaL i Priloien 2, 94 (1968). Vranceanu, G. [1] Leçons de Géométrie Différentielle, 3 volumes. Gauthier-Villars 1964. Wakakuwa, H. [1] On n-dimensional Riemannian spaces admitting some groups of motions of order less than n(n— 1)1 2. Tôhoku Math. J. 6, 121-134 (1954) (MR 16, 956). Wang, H.C. [1] Fins ler spaces with completely integrable equations of Killing. J. London Math. Soc. 22, 5-9 (1947) (MR 9, 206). [2] On invariant connections over a principal fibre bundle. Nagoya Math. J. 13, 1-19 (1958) (MR 21 *6001). [3] Two point homogeneous spaces. Ann. of Math. 55, 177-191 (1952) (MR 13, 863). [4] Closed manifolds with homogeneous complex structure. Amer. J. Math. 76, 1-32 (1954) (MR 16, 518). Wang, H.C., Yano, K. [1] A class of affinely connected spaces. Trans. Amer. Math. Soc. 80, 72-96 (1955) (MR 17, 407). Weber, W. C., Goldberg; S. I. [1] Conformal deformations of riemannian manifolds. Queen's paper in pure & applied Math., No. 16, Queen's Univ. Kingston, Ont. 1969 (MR 40 *1938). Weil, A.

[1] Introduction i l'Étude des Variétés Kahlériennes. Paris: Hermann 1958 (MR 22 *1921). Weinstein, A. [1] A fixed point theorem for positively curved manifolds. J. Math. Mech. 18, 149-153 (1968) (MR 37 *3478). [2] Symplectic structures on Banach manifolds. Bull. Amer. Math. Soc. 75, 1040-1041 (1969). Weyl, H. [1] Reine Infinitesimalgeometrie. Math. Z. 2, 384-411 (1918). [2] Zur Infinitesimalgeometrie; Einordnung der projektiven und konformen Auffassung. Giittinger Nachrichten (1912), 99-112. Wolt J. A. [1] Spaces of constant curvature. New York: McGraw-Hill 1967 (MR 36 *829). [2] Homogeneity and bounded isometries in manifolds of negative curvature. Illinois J. Math. 8, 14-18 (1964) (MR 29 *565).

180

Bibliography

[3] The geometry and structures of isotropy irreducible homogeneous spaces. Acta Math. 120, 59-148 (1968) (MR 36 *6549). [4] Growth of finitely generated solvable groups and curvature of riemannian manifolds. J. Differential Geometry 2,421-446 (1968) (MR 40 # 1939). [5] The automorphism group of a homogeneous almost complex manifold. Trans. Amer. Math. Soc. 144, 535-543 (1969) (MR 41 *956). Wright, E. [1] Thesis, Univ. of Notre Dame, Wu, H. [1] Normal families of holomorphic mappings. Acta Math. 119,193-233 (1967) (MR 37 #468). Yano, K. [1] On harmonic and Killing vector fields. Ann. of Math. 55, 38-45 (1952) (MR 13, 689). [2] On n-dimensional Riemannian spaces admitting a group of motions of order n(n — 1) + 1. Trans. Amer. Math. Soc. 74, 260-279 (1953) (MR 14,688). [3] The theory of Lie derivatives and its applications. Amsterdam: North-Holland, 1957 (MR 19,576). [4] Sur un théorème de M. Matsushima. Nagoya Math. J. 12, 147-150 (1957) (MR 20 *2476). [5] Harmonic and Killing vector fields in compact orientable manifolds Riemannian spaces with boundary. Ann. of Math. 69, 588-597 (1959) (MR 21 #3887). [6] Some integral formulas and their applications. Michigan Math. J. 5, 68-73 (1958) (MR 21 *1860), [7] Differential geometry on complex and almost complex spaces. New York: Pergamon Press 1965 (MR 32 #4635). [8] On Riemannian manifolds admitting an infinitesimal conformal transformations. Math. Z. 113,205-214 (1970) (MR 41 *6114). [9] Integral formulas in Riemannian geometry. New York: Marcel Dekker 1970. Yano, K., Bochner, S. [1] Curvature and Betti numbers. Ann. of Math. Studies, No. 32. Princeton Univ. Press 1953 (MR 15,989). Yano, K., Kobayashi, S. [1] Prolongations of tensor fields and connections to tangent bundles. J. Math. Soc. Japan 18, 194-210 (1966) (MR 33 # 1816); II. 236-246 (MR 34 #743); III. 19, 486-488 (1967) (MR 36 # 2084). Yano, K., Nagano, T. [1] Einstein spaces admitting a one-parameter group of conformal transformations. Ann. of Math. 69,451-461 (1959) (MR 21 *345). [2] Some theorems on projective and conformal transformations. Indag. Math. 19, 451-458 (1957) (MR 22 # 1861). [3] The de Rham decomposition, isometries and affine transformations in Riemannian spaces. Japan. J. Math. 29, 173-184 (1959) (MR 22 411341). Yano, K., Obata, M. [1] Conformal changes of Riemannian metrics. J. Differential Geometry 4, 53-72 (1970) (MR 41 # 6113). Yau, S. T. [1] On the fundamental group of compact manifolds of non-positive curvature. Ann. of Math. 93, 579-585 (1971) (MR 42 #8423).

Index

adjoint action 140 admissible coordinate system 1, 37 affine structure 35 affine transformation 122 — —, infinitesimal 42 almost complex structure 7 almost Hamiltonian structure 11 almost Hermitian structure 10 almost symplectic structure 11 ample 83 atlas 34 —, r-atlas 34 —, maximal (complete) 34 automorphism of a Cartan connection 128 — of a G-structure 2

Bergman kernel form 78 Bergman kernel function 78 Bergman metric 78 canonical form 141, 143 canonical line bundle 83 Cartan connection 127 — —, automorphism of a 128 characteristic class 67 characteristic number 67 chart 34 complete atlas 34 complete hyperbolic 81 complete vector field 46 conformal connection 136 conformal equivalence 9, 146 conformal structure 9, 142 — —, flat 36 conformal-symplectic structure 12 conformal-symplectic transformation 27 conformal transformation 143, 148 contact form 28 contact structure 28

contact transformation 29 — —, infinitesimal 29 degree of a G-structure 37 degree of a pseudogroup 36 degree of (compact) symmetry elliptic linear Lie algebra

55

4, 16

filtered Lie algebra 37 — — —, transitive 37 flat conformal structure 36 flat G-structure 35 flat projective structure 36 foliation 12 frame 36, 139 fundamental vector field 127, 140 1-atlas 34 —, maximal, complete 34 r-manifold 34 r-structure 34 G-structure 1, 33 —, degree of a 37 —, flat 35 —, integrable 1 —, prolongation of a 22 general type (algebraic manifold of) 87 graded Pe algebra 38 — — —, transitive 38 Hamiltonian structure 11 — —, almost 11 hyperbolic manifold 81 — —, complete 81 ILH-Lie group 23 infinite type (Lie algebra of) 4

181

Index

infinitesimal — affine transformation 42 — automorphism of a G-structure — contact transformation 29 — isometry 42 — symplectic transformation 11 integrable G-structure 1, 37 intrinsic pseudo-distance 81 isometry 39 —, infinitesimal 42 Killing vector field 42

residue 69 Riemann-Hurwitz relation

Lie pseudogroup 36

negative first Chern class 82 nonpositive first Chem class 103 order of a linear Lie algebra 4 projective connection

88

symplectic structure 11, 23 — —, almost 11 symplectic transformation 11, 25 — —, infinitesimal 11

maximal atlas 34 model space 34 Möbius space 133

parallelisable manifold

2

projective equivalence 145 projective structure 142 projective transformation 143 prolongation of a G-structure 22 — of a linear Lie algebra 4 — of a linear Lie group 19, 20 pseudogroup of transformations 34 — — —, degree of a 36 — — —, Lie 36 — —, transitive 34

13 136

transitive filtered Lie algebra — graded Lie algebra 38 — pseudogroup 34

very ample 78 volume element

6, 23

37

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