Transformation Groups in Differential Geometry (Classics in ...
Short Description
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 â 1, the ...
Description
Shoshichi Kobayashi
Transformation Groups in Differential Geometry Reprint of the 1972 Edition
Springer
Shoshichi Kobayashi Department of Mathematics, University of California
Berkeley, CA 947203840 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 3540586598 SpringerVerlag Berlin Heidelberg New York
Photograph by kind permission of George Bergman
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© SpringerVerlag 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 acidfree paper
Shoshichi Kobayashi
Transformation Groups in Differential Geometry
SpringerVerlag 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 3540058486 SpringerVerlag Berlin Heidelberg New York ISBN 0387058486 SpringerVerlag 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, reuse 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 SpringerVerlag Berlin Heidelberg 1972. Library of Congress Catalog Card Number 7280361. 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 Gstructure or that of pseudogroup 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 pseudogroups; 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 GStructures 1 1. GStructures 5 2. Examples of GStructures 3. Two Theorems on Differentiable Transformation Groups. • 13 16 4. Automorphisms of Compact Elliptic Structures 19 5. Prolongations of GStructures 23 6. Volume Elements and Symplectic Structures 28 7. Contact Structures 8. Pseudogroup Structures, GStructures 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 1Forms . • 90 4. Holomorphic Vector Fields on Kahler Manifolds . . . • 92 95 5. Compact EinsteinKahler 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 1Forms and Closed 2Forms 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 GStructures
1. GStructures
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 Gstructure on M we shall mean a differentiable subbundle P of L(M) with structure group G. There are very few general theorems on Gstructures. But we can ask a number of interesting questions on Gstructures, and they are often very difficult even for some specific G. It is therefore essential for the study of Gstructures to have familiarity with a number of examples. In general, when M and G are given, there may or may not exist a Gstructure 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 Gstructures on M are in a natural onetoone correspondence with the cross sections
M L(M)/G (see, for example, KobayashiNomizu [1, vol. 1; pp. 5758]). The socalled obstruction theory gives necessary algebraictopological conditions on M for the existence of a Gstructure (see, for example, Steenrod [1]). A Gstructure 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 Gstructure 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 GStructures
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 Gstructure 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 Gstructure 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. GStructures
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 Gstructures 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 Gstructure P onto the Gstructure P. If M=M' and P = P', then an isomorphism f is called an automorphism of the Gstructure P. A vector field X on M is called an infinitesimal automorphism of a Gstructure P if it generates a local 1parameter group of automorphisms of P. As in Proposition 1.1, we consider those Gstructuresdefined 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 Gstructure. 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:
c°
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 GStructures
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 kth 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 GStructures
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 Gstructure P =
R" x G on Rn. For a survey on Gstructures, 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 Gstructure 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 Gstructures 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 areameasure structures (Brickell [1]), nor pseudoconformal structures of real hypersurfaces in complex manifolds (MorimotoNagano [1], Tanaka [3]) although automorphism groups of these structures are usually Lie groups. 2. Examples of GStructures 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 Gstructure 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 Gstructure 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 GStructures
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 onetoone correspondence with the volume elements of M, L e., the nforms 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 GStructures
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 onetoone 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; 0structure 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, KobayashiNomizu [1, vol. 2; p. 321]. The theorem of NewlanderNirenberg 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 GStructures
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(3/3x1)=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 skewsymmetric matrices, we have t.i k = — tf k . Hence, 4k=tftjtjt
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 onetoone 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 onetoone correspondence between 
9
2. Examples of GStructures
the pseudoRiemannian 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 pseudoRiemannian metric has vanishing curvature. It should be remarked that, although every paraconipact manifold admits a Riemannian metric, it may not in general admit a pseudoRiemannian metric of signature q for q #0, n. For automorphism of pseudoRiemannian 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 indexfree 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 onetoone correspondence with the CO(n)structure on M. A conformal
1. Automorphisms of GStructures
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 socalled 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 mdimensional 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 GStructures
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 2mdimensional manifold M are in a natural onetoone correspondence with the 2forms 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 2form 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 KobayashiNomizu [1, vol. 1; p. 149]). A diffeomorphismf of M onto itself is an automorphism of the symplectic structure defined by a 2form 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 GStructures
12
A CSp (m; R)structure is called a conformalsymplectic structure. For conformalsymplectic 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 pdimensional 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 onetoone correspondence with the pdimensional distributions on M, i. e., the fields of pdimensional 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 pdimensional distribution. In other words, a GL(p, q; R)structure is integrable if and only if the corresponding pdimensional distribution is involutive, (see Frobenius theorem). An integrable GL(p, q; R)structure is known as a foliation with pdimensional leaves. If a GL (p, q ; 10structure 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 pdimensional 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 onetoone 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 onetoone 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 1parameter groups (p i = exp t X of transformations of M such that (A0:5. If the set S generates a finitedimensional 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 1parameter subgroup of generated by X while we denote by exp t X the 1parameter 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 GStructures
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(n1)+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)=1n(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 ndimensional 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 —(n1)=1n(n1)+1—(n1)=1(n1)(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 1dimensional subspace of Rn invariant. If n =7, then either 5 = SO (n —1) leaving a 1dimensional 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(n1) 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 MontgomerySamelson [1] by a topological method. We indicate an algebraic proof. First, assume that the action of 5 on Rn is reducible with a pdimensional invariant subspace W. Then it leaves an (n —p)dimensional orthogonal complement Rn13 invariant. Hence, dim b_dim 0(p)+dim 0(n —p)=1p(p —1)+1(n —p)(n—p1). 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 semisimple 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 1dimensional 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 n1, 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(n1)/ '
6 leaves m' elementwise fixed, so that [I), nti =0. We shall show that either [m', m"] =0
or [m', m"] =m".
Fix a nonzero 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" =n1 or the whole space m" so that [m', m"]=0. We assume dim [m', nt"] =n1. 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, 2.1n, 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 6invariant 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 semisimple. 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 5invariant 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 ndimensional Riemannian manifolds M such that 3(M) contains a closed subgroup of dimension In(n1)+ 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 IVI1 and (6 is the direct product of the group of translations on R and the group of proper motions of R". To find a nonsimply 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 nonflat 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, n1) 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 Stinvariant 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 1dimensional 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 1dimensional 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 KobayashiNomizu [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 semidirect 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 ndimensional Riemannian manifold with n >4 such that its group 3 (M) of isometries contains a closed connected subgroup lowing:
of dimension 5 n(n1)+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 0invariant unit vector field X and admits a 0invariant 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) (semidirect 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 fourdimensional group of isometries acts on a 3dimensional Riemannian manifold, the action is transitive. E. Cartan ([8; pp. 293306]) has classified all such groups together with their actions.
55
4. Riemannian Manifolds with Little Isometries
In the 4dimensional case, difficulties arise from the fact that SO(4) is not simple. The 4dimensional 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 ndimensional Riemannian manifold, then 3(M) contains no compact subgroups of dimension r for 1(n — k)(n—k+1)+1k(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 1parameter 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 decktransformations which is a discrete group, the 1parameter family h, of transformations must commute with the decktrans:
58
II. Isometrics of Riemannian Manifolds
formations elementwise. Since the decktransformations 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 nonpositive sectional curvature, this geodesic exists and is unique (see, for example, KobayashiNomizu [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 nondifferentiable versions of some of the results in this section, see Busemann [2 ]. The following result of AtiyahHirzebruch [1] implies that a compact manifold M with nonzero Âgenus 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 Âgenus 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 nonempty, 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, autoparallel submanifolds (see KobayashiNomizu [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 arclength 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 oneparameter 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 1parameter subgroup of 0 generated by X leaves c fixed pointwise. Since 0 is connected and is generated by these 1parameter 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 1parameter 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 ( skewsymmetric) 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 1parameter 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 eigenvalues + 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 eigenvalues — 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 KobayashiNomizu [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 sneighborhood 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 1parameter 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 1parameter 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 orientationpreserving, then f has a fixed point. (2) If n =dim M is odd and f is orientationreversing, 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 nonnegative 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 1parameter family of curves cs , —eResf (1Vi).
pgi
Proof The first formula we are going to prove is the following:
( 1)
tdf(tflEVX)=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 1form on MZero (X) by
( Y) = g (X, Y)/g (X, X)
for Ye T ( MZero (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 +hi2+DX) tchk 1—tdtk
1 —t
and
1
ixn=ixi(to+vx) 1t +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/mIr 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 MZero(X).
Let N denote the 6neighborhood 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 eneighborhood 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 MZero (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/JFt 2 tfrA(dt/4 2 +•••, 1tdtk
the problem is further reduced to that of calculating a* (4/ A (d Or 1.
II. Isometrics of Riemannian Manifolds
72
Let be the 1form 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 AdyiEk 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 skewsymmetric nondegenerate 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 nondegenerate 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)41 ) 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, 321323 ]. )
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)411)=
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 dth 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 2form —L E g Œ, .de A d? 2ni may be considered as a 2form on M. From the definition of the Bergman metric, it is clear that this 2form 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 nonsingular 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 KodairaSpencer [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 starlike 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 nfold 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 RiemannHurwitz 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 RiemannHurwitz relation. The degree of ramification of f at a is equal to m1. If we denote by n the order of 15, then the sorbit through a consists of n/m points. The sum of the degrees of ramification of f at these points on the sorbit 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
(2p2)/n >2
= 3. >1.. =2 > =T1 =>1.
ri 3(p— I) ir 4(p1) 71. 6(p1) 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
(2p2)/n
>3 >3 >3 >4 >4 >3
>+ > =w1 =12 1
8(p1) n12(p1) n24(p1) n 20(p 1) 7240(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 1form 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 BorelNarasimhan [1]. For more results on automorphisms of compact Riemann surfaces of genus 2, see Macbeath [1], LehnerNewman [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 upperhalf 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 1Forms If Z is a holomorphic vector field and co is a holomorphic 1form 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 1forms on M. Define (Zeb; co(Z)=0 for all co EV.°)
91
3. Holomorphic Vector Fields and Holomorphic 1Forms
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 1form 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 antiholomorphic 1forms. 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) 1forms. Then
Let 21 be the space of cobounding 1forms, i.e., 1forms of the type d f. Then
Let
g. 01. ° = {coeV. ° ; co(Z)=0for 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 nonempty 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 nonpositive, nonnegative 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 Taz—a +Evpcadzfico v"z=
vp
a a? Taz,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 1form 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=HCFd"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 1form, 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 1forms 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 HodgeKodaira 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 EinsteinKOhler 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 1form, 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 1form 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 EinsteinKHhler Manifolds In this section we shall prove the following result of Matsushima [I]. Theorem 5.1. Let M be a compact EinsteinKahler 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 =11C+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 EinsteinKahler 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 nonhomogeneous compact Einstein manifold. In this connection, see Berger [2] for a survey on Einstein manifolds and Aubin [1] for a construction of certain EinsteinKahler 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 1forms 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=vpR
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
E973v,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 1form. 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 nonnegative 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"91(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 4harmonic if di:, 9=0. Theorem 7.1. Let M be a compact Keihler manifold and denote by the space of 4harmonic (p, q)forms and by OP the sheaf of germs of holomorphic pforms 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 TaF , 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 vpcce • 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 2nform 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 1forms 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 1forms 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 semide 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)113 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 semidefinite 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 semidefinite, 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 i1harmonic 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 1forms 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 1form 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 nform 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 rpFt 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 nonnegative.) Since Z(f) , I1Z112 is positive almost everywhere, we conclude that 9 q.e.d. vanishes identically on M. Since a holomorphic nform 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, nonnegative (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 1parameter 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 pform 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 kdimensional 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(n1)dimensional homology class represented by a hyperplane. Lemma 1. Let M and Z be as above. If p is a nonzero holomorphic pform on M, then there exist rdimensional 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 kdimensional 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 41)=[(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 pdimensional 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 pdimensional 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 pforms 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 ksheeted 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 RiemannRochHirzebruch 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 nonnegative first Chem class. )]
12. Holomorphic Vector Fields and Characteristic Numbers Let M be an ndimensional 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 nondegenerate 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 0form 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 nondegenerate 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 BaumBott [1]. For a different proof, see AtiyahSinger [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 ndimensional 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 KobayashiNomizu [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 ndimensional 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 ndimensional 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 skewsymmetric bilinear mapping V x V—) V Let X: V—) V be a linear transformation and 91 = et x be the 1parameter 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 nonvanishing components of T Since Tik is skewsymmetric 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 nonsingular. 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 = A12 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 nonsingular. 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 1form 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 1form 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 ndimensional 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 ) = 901 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 KobayashiNomizu [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 1dimensional Euclidean space. Let M be a complete Riemannian manifold and AI be its universal covering space. Let ft4 = 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], YanoNagano [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 KobayashiNomizu [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 ndimensional 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 ndimensional 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 ndimensional real projective space and 20 an isotropy subgroup of 2 so that 2/20 is the projective space. Let M be an ndimensional 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 ndimensional 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 ndimensional 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 (KobayashiNomizu [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 1form 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 I0valued. 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 KobayashiNomizu [1; Chapter III, §§ 23].) 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 Ivalued 2form on P. For instance, if a) is an affine connection, then Q is a 2form 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 Rncomponent 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 KobayashiNomizu [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 ==gi+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 I0component w 0 + w 1 + • • • + Wk, restricted to each fibre of P, is the MaurerCartan 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 MaurerCartan 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 Ivalued 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 20bundle 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 gocomponent 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 KobayashiNagano [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=91+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 KillingCartan 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 KillingCartan 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 nondegenerate, 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 semisimple 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, Œ (E)=: E. By the definition of a, E ego .
The graded Lie algebras of Proposition 4.1 have been classified by KobayashiNagano [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 nvector,
21 = { (
y I 0 1 ) ; c: row nvector } .
The graded Lie algebra I= g_ ii + go +g1 with this .2/20 is given by
1=5I(n+1; R),
91= {
Co o)
v },
go= I(
A 0 ); trace A + a=0}, 0 a
where y is a column nvector, The element Eeg o is given by
E=
lain 0\
k o bi '
g1=
{(0 00» ,
is a row nvector, 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 ndimensional vector space of column nvectors and 1/* be the dual space consisting of row nvectors. Then I= 17 1gI(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 MaurerCartan form co of PGL(n; R) as follows:
co =Eal ei +Ea4 el +Ecoi ei . Then the MaurerCartan 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 ndimensional Möbius space Sn, first geometrically and then grouptheoretically. 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 ndimensional Möbius space. It is diffeomorphic to the nsphere 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}, (a1 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 nvector, Eego is given by
is a row nvector 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 MaurerCartan form co of 0 (n +1, 1) as follows:
co=Ecoi ei +Eco.Vg +Ecoi ej, where (coi) is co(n)valued. Then the MaurerCartan 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 20bundle over M. Given a g_ 1 valued 1form co_ 1 and govalued 1form coo , we consider the question whether there exists a natural 91 valued 1form 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 gocomponent of A; (b')
W ar (co _ 1 + °k)= (ad a  1 ) (w_ 1 + coo)
for every a e 20 ,
where ad a1 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 20bundle over a manifold M of dimension n (.. 3 for Ex. 4.2). Given a g_ 1 valued 1form w_ 1 =(cd) and a govalued 1form 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 COi
=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 n2 2(n1) '
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) COŒ a
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=E14 / cok A co';
(2)E
Qi = 0, or equivalently, Kiki+ Kku+ Kuk= 0, where Q; =EAK 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 (2n4) 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 ndimensional 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 rjet at 0 if they have the same partial derivatives up to order r at O. The rjet 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 rjet f0 (f) at 0 is called an rframe at x = f (0). Clearly, a 1frame is an ordinary linear frame. The set of rframes 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 rframes 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 1form 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 1form 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 2frame 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 setup 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
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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 2jets (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 2jet 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 onetoone 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 2frames of M into the bundle P2 (M') of 2frames 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 Gstructure 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 Gstructure. 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 2jet j (f) can be considered as a 2frame of 2/20 The set of all 2frames thus obtained defines a projective structure atf(o). on 2/20 which can be identified with the bundle L over 2/20 . Then the MaurerCartan form co =(co1 ; coi ; ao of .2 defined in Example 4.1 defines the normal projective connection of this projective structure. The MaurerCartan 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 grouptheoretic 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 nvector. 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 MaurerCartan 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 MaurerCartan 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 1parameter group of projective transformations of M lifts to alparameter 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 fibretransitive 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 MaurerCartan 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 MaurerCartan 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 1form as well as a Yvalued 1form. Two torsionfree affine connections of M defined by govalued 1forms 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 KobayashiOchiai [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 Yvalued form (i.e., a form whose values are column nvectors) and p as a Y*valued function (i.e., a function whose values are row nvectors), 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 govalued 1forms 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 2frames over an ndimensional 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 onetoone correspondence with the torsionfree affine connections of M. (2) The cross sections M—> P2 (M)/ 130 are in onetoone 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. .:(01 )
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 onetoone 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 1form 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], YanoNagano [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 LelongFerrand [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 GoldbergKobayashi [1], Kulkarni [1], LedgerObata [1], Lichnerowicz [1], Obata [37, 9], SuyamaTsukamoto [1], Tanaka [1], Tashiro [2], TashiroMiyashita [1, 2], WeberGoldberg [1], Yano [3, 8], YanoNagano [2], YanoObata [1], and references therein. A list of papers which had given partial results toward Theorem 7.5 can be found in YanoObata [1].
Appendices
1. Reductions of 1Forms and Closed 2Forms
For the discussion of contact structures and symplectic structures, it is important to know the simplest possible expressions for 1forms and closed 2forms. Let a) be a 1form 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 1form 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 1form 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 2k1 d x 2k for p = 2 k, w=x dx2+x3 dx4+• +x2k1 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 1form 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 1Forms and Closed 2Forms
151
da) is a skewsymmetric 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=n2k+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) 1form 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(n1) = 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 1form 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 1Forms and Closed 2Forms
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 1form 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 21c1 = 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 1form which does not involve dy l , , dy k. As we have explained before Lemma 4, ik may be considered as a 1form 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 2k1 = 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 2form 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 2form 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 1form 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 P1 dx2 P Or
co=xidx2 +.±x 2 P1 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 +
Œ ill +
•
)
and
E yva•iP)=E(vp
Vg By adding these two equalities and integrating the resulting equality over M, we obtain E (Rap OE r711 + 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 1form corresponding to X under the duality defined by the Riemannian metric. If we denote by A the Laplacian, then A c is a 1form. 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"DpieFERzpŒijfl+EV0 Œ.Vs zt} dv=0. 3. Laplacians in Local Coordinates Let co be a pform 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 (p1)!
. 12 ...1„. dx f2 A
dxiP.
In particular, let co be a 1form 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 xI.
For a similar expression for a pform 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
... 41 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) difl =(—Va VacopcigFR 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, p1), (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 =1113 +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.
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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 —, ratlas 34 —, maximal (complete) 34 automorphism of a Cartan connection 128 — of a Gstructure 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 conformalsymplectic structure 12 conformalsymplectic transformation 27 conformal transformation 143, 148 contact form 28 contact structure 28
contact transformation 29 — —, infinitesimal 29 degree of a Gstructure 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 Gstructure 35 flat projective structure 36 foliation 12 frame 36, 139 fundamental vector field 127, 140 1atlas 34 —, maximal, complete 34 rmanifold 34 rstructure 34 Gstructure 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 ILHLie group 23 infinite type (Lie algebra of) 4
181
Index
infinitesimal — affine transformation 42 — automorphism of a Gstructure — contact transformation 29 — isometry 42 — symplectic transformation 11 integrable Gstructure 1, 37 intrinsic pseudodistance 81 isometry 39 —, infinitesimal 42 Killing vector field 42
residue 69 RiemannHurwitz 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 Gstructure 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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