Riemannian Geometry of Contact and Symplectic Manifolds (Progress in mathematics )

Progress in Ntahematics

David E. Blair

Riemannian Geometry of Contact and Symplectic Manifolds

Birkhauser

Progress in Mathematics Volume 203

Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

David E. Blair

Riemannian Geometry of Contact and Symplectic Manifolds

Birkhauser Boston Basel Berlin

David E. Blair Department of Mathematics Michigan State University East Lansing, MI 48824-1027 U.S.A.

Library of Congress Cataloging-in-Publication Data Blair, David E., 1940 Riemannian geometry of contact and symplectic manifolds / David E. Blair. p. cm. - (Progress in mathematics (Boston, Mass.)) Includes bibliographical references and index. ISBN 0-8176-4261-7 (alk. paper) - ISBN 3-7643-4261-7 (alk. paper) 1. Contact manifolds. 2. Symplectic manifolds. 3. Geometry, Riemannian. I. Title. II. Series.

QA614.3.B53 2001 2001052707

516.3'73-dc2l

CIP

AMS Subject Classifications: 53B35, 53C15, 53C25, 53C26, 53C40, 53C42, 53C55, 53C56, 53D05, 53D10, 53D12, 53D15, 53D22, 53D25, 53D35, 58E11 Printed on acid-free paper ©2002 Birkhauser Boston

Birkhauser

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-8176-4261-7

SPIN 10845070

ISBN 3-7643-4261-7

Reformatted from author's files by TXniques, Inc., Cambridge, MA Printed and bound by Hamilton Printing Company, Rensselaer, NY Printed in the United States of America

987654321

To Rebecca in appreciation of all her love and support

Contents

1

Symplectic Manifolds 1.1 1.2 1.3 1.4

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7 9

Principal Sl-bundles 2.1 2.2

3

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Definitions and examples . . . . Lagrangian submanifolds The Darboux-Weinstein theorems Symplectomorphisms . . .

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The set of principal Sl-bundles as a group . . Connections on a principal bundle .

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Contact Manifolds 3.1 3.2

Definitions Examples 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.2.9

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.. .. .. 118n+1 x P118n .. Men+1 C R2n+2 with TTM2n+1 fl {0} = 0 .. Tl M, T1M .. .

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............................. ........................ ...

T*M x1[8 T3 T5

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Overtwisted contact structures . Contact circles . . . 3.3 The Boothby-Wang fibration . .. 3.4 The Weinstein conjecture . . . .

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17 20 20 21 21 22 22 23 23 24 25 26 28

Contents

viii

31

4 Associated Metrics 4.1 Almost complex and almost contact structures . 4.2 Polarization and associated metrics . . . 4.3 Polarization of metrics as a projection . . . 4.3.1 Some linear algebra . . .

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4.3.2 4.4 4.5

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R2n+1

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M2n+1 C M2n+2 almost complex . . . S5 C S6 . . .

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The Boothby-Wang fibration M2n X R

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Parallelizable manifolds

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31

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Action of symplectic and contact transformations Examples of almost contact metric manifolds . 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6

5

Results on the set A .

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34 38 39 41 45 48 48 49 50 52 53 53

Integral Submanifolds and Contact Transformations 5.1 5.2 5.3

. . . Integral submanifolds . . . . Contact transformations . . Examples of integral submanifolds . . . . . 5.3.1 Sn C Stn+r . . . . . 5.3.2 T2 C S5 5.3.3 Legendre curves and Whitney spheres . 5.3.4 Lift of a Lagrangian submanifold . . .

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55

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6 Sasakian and Cosymplectic Manifolds 6.1 6.2 6.3

Normal almost contact structures . . . The tensor field h . Definition of a Sasakian manifold .

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. . . . . . . . . CR-manifolds Cosymplectic manifolds and remarks on the Sasakian definition . . . . . 6.6 Products of almost contact manifolds . . . . . . . . . 6.7 Examples .

6.4 6.5

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6.7.1 6.7.2 6.7.3 6.7.4 6.7.5

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Brieskorn manifolds 6.8 Topology . .

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7 Curvature of Contact Metric Manifolds 7.1 7.2

. Basic curvature properties . Curvature of contact metric manifolds .

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. . . . Principal circle bundles A non-normal almost contact structure on S5

M2n+l C M2n+2

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81 81 81 83 85 85

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91 91 95

Contents 7.3 7.4

7.5

. . 0-sectional curvature . . . Examples of Sasakian space forms

7.4.1 7.4.2

Sen.+1

7.4.3

B"-

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xR

Locally 0-symmetric spaces

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110 114

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114

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114 115

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115

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8 Submanifolds of Kahler and Sasakian Manifolds 8.1 8.2 8.3

Invariant submanifolds . . . Lagrangian and integral submanifolds Legendre curves . . . . . . .

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121

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121

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9 Tangent Bundles and Tangent Sphere Bundles 9.1 9.2 9.3 9.4

Tangent bundles . . . . Tangent sphere bundles . . Geometry of vector bundles Normal bundles . . .

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137

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137 142 148 150

10 Curvature Functionals on Spaces of Associated Metrics 10.1 Introduction to critical metric problems 10.2 The *-scalar curvature . . 10.3 The integral of Ric(h) . . . . 10.4 The Webster scalar curvature 10.5 A gauge invariant . . . . 10.6 The Abbena metric as a critical point .

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157

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157 162 166 170 173 174

11 Negative c-sectional Curvature

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11.1 Special Directions in the contact subbundle 11.2 Anosov flows . . . 11.3 Conformally Anosov flows .

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177 178 184

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12 Complex Contact Manifolds

189

12.1 Complex contact manifolds and associated metrics 12.2 Examples of complex contact manifolds . . . 12.2.1 Complex Heisenberg group . . . . . 12.2.2 Odd-dimensional complex projective space 12.2.3 Twistor spaces . . . . . . . . 12.2.4 The complex Boothby-Wang fibration . 12.2.5 3-dimensional homogeneous examples . . .

12.2.6 C"`+1 X CP'-(16)

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200

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12.3 Normality of complex contact manifolds 12.4 GH-sectional curvature . . . . .

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189 193 193 194 196 198 200

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202 204

x

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12.5 The set of associated metrics and integral functionals . . 12.6 Holomorphic Legendre curves . . . . . . . . . . 12.7 The Calabi (Veronese) imbeddings as integral submanifolds .

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206 209

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212

13 3-Sasakian Manifolds 13.1 3-Sasakian manifolds 13.2 Integral submanifolds

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215 223

Bibliography

227

Subject Index

253

Author Index

257

Preface

The author's lectures, "Contact Manifolds in Riemannian Geometry," volume 509 (1976), in the Springer-Verlag Lecture Notes in Mathematics series have

been out of print for some time and it seems appropriate that an expanded version of this material should become available. The present text deals with the Riemannian geometry of both symplectic and contact manifolds, although the book is more contact than symplectic. This work is based on the recent research of the author, his students, colleagues, and other scholars, the author's graduate courses at Michigan State University and the earlier lecture notes. Chapter 1 presents the general theory of symplectic manifolds. Principal circle bundles are then discussed in Chapter 2 as a prelude to the BoothbyWang fibration of a compact regular contact manifold in Chapter 3, which deals with the general theory of contact manifolds. Chapter 4 focuses on Riemannian metrics associated to symplectic and contact structures. Chapter 5 is devoted to integral submanifolds of the contact subbundle. In Chapter 6 we discuss the normality of almost contact structures, Sasakian manifolds, Kcontact manifolds, the relation of contact metric structures and CR-structures, and cosymplectic structures. Chapter 7 deals with the important study of the curvature of a contact metric manifold. In Chapter 8 we give a selection of results on submanifolds of Kahler and Sasakian manifolds, including an illus-

tration of the technique of A. Ros in a theorem of F. Urbano on compact minimal Lagrangian submanifolds in CP'z. Chapter 9 discusses the symplectic structure of tangent bundles, contact structure of tangent sphere bundles, general vector bundles and normal bundles of Lagrangian and integrals submanifolds giving rise to new examples of symplectic and contact manifolds.

xii

Preface

In Chapter 10 we study a number of curvature functionals on spaces of associated metrics and their critical point conditions; we show also that in the symplectic case, the "total scalar curvature" is a symplectic invariant and in the contact case is a natural functional whose critical points are the metrics for which the characteristic vector field generates isometries. In the presence of a certain amount of negative curvature, special directions appear in the contact subbundle; we discuss these and their relations to Anosov and conformally Anosov flows in Chapter 11. Chapter 12 deals with the subject of complex contact manifolds. We conclude with a brief treatment of 3-Sasakian manifolds in Chapter 13. The text attempts to strike a balance between giving detailed proofs of basic properties, which will be instructive to the reader, and stating many results whose proofs would take us too far afield. It has been impossible, however, to be encyclopedic and include everything, so that unfortunately some important topics have been omitted or covered only briefly. An extensive bibliography is given.

It is the author's hope that the reader will find this both a good introduction to the Riemannian geometry of contact and symplectic manifolds and a useful reference to recent research in the area.

The author expresses his appreciation to C. Baikoussis, B.-Y. Chen, D. Chinea, T. Draghici, B. Foreman, Th. Koufogiorgos, Y.-H. P. Pang and D. Perrone for reading parts of the manuscript and offering valuable suggestions. The author also expresses his appreciation to Ann Kostant of Birkhauser and to Elizabeth Loew of TFYniques for her kind assistance in the production of this book.

David E. Blair

October, 2001

1

Symplectic Manifolds

1.1

Definitions and examples

To set the stage for our development, we begin this book with a treatment of the basic features of symplectic geometry. By a symplectic manifold we mean an even-dimensional differentiable (C°°) manifold Men together with a global

2-form St which is closed and of maximal rank, i.e., dSl = 0, Stn 0 0. By a symplectomorphism f : (M,, Sll) -> (M2, Q2) we mean a diffeomorphism f : M1 -> M2 such that f *Q2 = Q1. Before continuing with symplectic manifolds, we present some basic linear algebra. On a vector space V2n, if St E A2 V with rkQ = 2n, then there exist B1, ... 02n E V*, linearly independent and such that

cl =

01

A

02

+ 93 A 04

+... +. 92n-1 A 02n.

To see this, for a basis {wi} of V* write

St=>aijw2Aw3 =cw1A i<j

w3, ..., w2n 2jw + terms in

A 1<j

2<j

= w1 A 01 + w2 A 02 + terms in where 02 involves only W3, ..., In and /31 = awe +,33 and 03 involves only w3, ..., w2n. Therefore 2w2n

1

1

a

a

St=W1A/31+-/31A02_ _ (w1 -

1,32) a

03n02+terms inW3,...,W2n

n /31 + terms in W3,

,W2n

2

1. Symplectic Manifolds

which is of the form 01 A 02 +c1 where 521 involves only w3 ..., w2n Now repeat the process for 521.

We shall often choose the labeling such that Q = 0' A 0n+1 + ... + 0" A 02n.

As a corollary we see that there exists a basis {ei, en+i} of V2n such that SZ(ei, en+j) = Si i, j = 1, ..., n. A change of basis which leaves invariant the normal form S2 = E71 0' A 0n+i is given by a symplectic matrix, i.e.,

D -C T if and only if (C D) - \ -B A ) AB

_

(C D)T ( 10)(C D) - ( 10)' In particular the structural group of the tangent bundle of a symplectic manifold is reducible to Sp(2n, IR). Further, using the fact that Men may be given a Riemannian metric and is orientable, the structural group is reducible to SO(2n) and hence in turn to U(n). Thus in particular Men carries an almost complex structure; this will be discussed in greater detail below and in Chapter 4. The name symplectic is due to H. Weyl [1939, p. 165] changing the Latin com/plex to the Greek sym/plectic. Two canonical examples of symplectic manifolds are the following:

yn) admits the symplectic form 1. R2n with coordinates (xl xn yl 52 = E dxi A dyi. The classical theorem of Darboux states that on any symplectic manifold there exist local coordinates with respect to which the symplectic form can be written in this way. We will give a modern proof of this result in Section 1.3. 2. Let M be a differentiable manifold; then its cotangent bundle T*M has a natural symplectic structure. For z E T*M and V E TZT*M define often called the Liouville form, by /3(V)z = z(7r*V) where a 1-form 7: T*M -- M is the projection map. If x1, ..., xn are local coordinates on M, qi = xi o 7r and fibre coordinates pl,.., pn form local coordinates on T*M; /3 then has the local expression E' l pidgi. The natural symplectic structure on T*M is given by S2 = -do.

The reader may recognize the second example from classical mechanics; in-

deed the cotangent bundle of the configuration space may be thought of as the phase space of a dynamical system and we may obtain Hamilton's equations of motion as follows. Let H be a real-valued function on a symplectic manifold (M, S2) and define a vector field XH by SZ(XH, Y) = YH. XH is

1.1 Definitions and examples

3

called the Hamiltonian vector field generated by H. Two basic properties of XH are £XH52 = 0, £ being Lie differentiation, and XHH = 0. In fact the classical Poisson bracket is {fl, f2} = 52(X f2, X f,) = X f, f2. In local coordiqn, pi pn) given by the Darboux Theorem, 52 = E dq' A dp' nates (ql

and XH = E ( - a4 api + P a Thus the differential equations for the integral curves of XH are

aH

pi

= - aq2 ,

i

q

aH = api

Hamilton's equations of motion. Before giving further examples, we mention the relationship with Riemannian geometry which will become central for our study, viz. "associated metrics". Given a symplectic manifold (M, 52), there exist a Riemannian metric g and an almost complex structure J such that 52(X, Y) = g(X, JY). In fact

we shall see in Chapter 4 that there are many such metrics, g and J being created simultaneously by polarization.

On the other hand, given an almost complex structure J, i.e., a tensor field J of type (1,1) such that J2 = -I, on an almost complex manifold, a Riemannian metric is said to be Hermitian if g(JX, JY) = g(X, Y) and the pair (J, g) is called an almost Hermitian structure. Now defining a 2-form ft by ft(X, Y) = g(X, JY), Q is called the fundamental 2-form of the almost Hermitian structure. If dS2 = 0, the almost Hermitian structure is said to be almost Kahler, whereas the structure is Kahler if J or equivalently Sl is parallel with respect to the Levi-Civita connection of g. Thus associated metrics can be thought of as almost Kahler structures whose fundamental 2-form is the given symplectic form. Since dQ = 0, [Q] E H2(M, R), [fl] denoting the deRham cohomology class determined by Q. Using an associated metric, bQ = 0 and hence SZ is harmonic.

To see this, use the fact that an almost Kahler manifold is quasi-Kahler, i.e.,

(VkJip)JJ = (V JJ,)Jkr, sum on the indices i and k and use J2 = -I. Also [52]x` = [52n] E H2n(M, R). In particular for M compact the following are two necessary conditions for the existence of a symplectic structure.

i) M carries an almost complex structure. ii) There exists an element w E H2 (M, R) such that wn

0.

Thus, for example, from i) S4 is not symplectic and from ii) S6 is not symplectic.

Gromov in his thesis proved that if M is an open manifold, then i) implies the existence of a 1-form w such that dw is symplectic (see A. Haefliger [1971, p. 133]). Also Kahler manifolds are symplectic so there are plenty of compact

4

1.

Symplectic Manifolds

ones, e.g., complex projective space, S2 x S2, algebraic varieties, etc. Note that the even-dimensional Betti numbers of a compact almost Kahler manifold are non-zero. It is also well known that the odd-dimensional Betti numbers of a

compact Kahler manifold are even, but this is not true in the almost Kahler case. We now give two descriptions of an example of Thurston [1976] of a compact symplectic manifold with no Kahler structure; this manifold is known as the Thurston manifold or as the Kodaira-Thurston manifold (Kodaira [1964]).

Briefly, first take the product of a torus T2, as a unit square with opposite sides identified, and an interval and glue the ends together by the diffeomorphism of T2 given by the matrix (

0

1

) . This gives a compact 3-manifold

whose first Betti number is 2. Now taking the product with S' we have a 4-manifold M with first Betti number 3 and hence M cannot have a Kahler structure. Let 8,, 92 be coordinates on T2; then d&1 Ad02 exists after the twisting on the 3-manifold. Thus if 01 is the coordinate on the interval and 02 the coordinate on the final circle, dOl A d02 + dq 1 A d02 is a symplectic form. A second version of this example was given by E. Abbena [1984] who also

gave a natural associated metric for this symplectic structure, computed its curvature and showed by using harmonic forms that the first Betti number was 3.

Let G be the closed connected subgroup of GL(4, C) defined by 1

a12

a13

0

0

1

a23

0

0

0

1

0

0

0

0

a12, a13, a23, a E II8

,

e27ria

i.e., G is the product of the Heisenberg group and S1. Let I' be the discrete subgroup of G with integer entries and M = G/I'. Denote by x, y, z, t coordinates on G, say for A E G, x(A) = a12, y(A) = a23, z(A) = a13, t(A) = a. If LB is left translation by B E G, LBdx = dx, LBdy = dy, LB(dz - xdy) = dz - xdy, LBdt = dt. In particular these forms are invariant under the action of F; let 7r : G -* M, then there exist 1-forms al, a2, a3i a4 on M such that dx = 7r*al, dy = 7r*a2, dz - xdy = 7r*a3i dt = 7r*a4. Setting Q = a4 A a1 + a2 A a3 we see that S2 A Sl 0 and dQ = 0 on M giving M a symplectic structure. The vector fields el

a

a'x

a + a e2__ay x

a e3=-, e4=-a

az' at are dual to dx, dy, dz - xdy, dt and are left invariant. Moreover {ei} is orthonormal with respect to the left invariant metric on G given by ds2

= dx2 + dy 2 + (dz

- xdy)2 + dt2.

1.2 Lagrangian submanifolds

On M the corresponding metric is g =

5

ai ® ai and is called the Abbena

metric. Moreover M carries an almost complex structure defined by

Jel = e4, Je2 = -e3, Je3 = e2, Je4 = -e1.

Then noting that 52(X, Y) = g(X, JY), we see that g is an associated metric. At the time of Thurston's example there was no known example of a compact

symplectic manifold with no Kahler structure. Since the appearance of this example there have been many others, e.g., Cordero, Fernandez and Gray [1986], Cordero, Fernandez and de Leon [1985], Gompf [1994], Holubowicz and Mozgawa [1998], Jelonek [1996], McCarthy and Wolfson [1994], McDuff [1984], Watson [1983].

1.2

Lagrangian submanifolds

Let t : L --> M2n be an immersion into a symplectic manifold (M2n, 52). We say that L is a Lagrangian submanifold if the dimension of L is n and t*52 = 0. Two simple examples are the following. 1. The fibres of the cotangent bundle T*M as discussed in the previous section are Lagrangian submanifolds with respect to the symplectic structure dpi. Also suppose that 0 is a section of T*M. Then 0*0 = as a 1-form on M; in particular

(0*0)(X )m = 0(0*X)0(m) = c(m)(i*O*X) _ q(X)m. 0*(-d,Q) = -do so that a section : M --> T*M is a Lagrangian submanifold if and only if 0 is closed. When 0 is exact, say dS, S is said to be a generating function for the submanifold. Therefore O*52 =

2. Let (M1, 521) and (M2, 522) be symplectic manifolds and f : M1 --i M2 a diffeomorphism. Then (MI, 521) x (M2, -p2) is symplectic, say M = M1 x M2 with projections 71 and 72 and SZ = 71Q1-7r2522. Let r f denote

the graph of f. Then f is a symplectomorphism if and only if I' f is a Lagrangian submanifold of (M, S2).

There are several difficulties in studying Lagrangian submanifolds. First of all, since t*S2 = 0, there is no induced structure, so in a sense the geometry is transverse to the submanifold. From the standpoint of submanifold theory the codimension is high so that theory is more complicated. Lagrangian submanifolds are very abundant. For example, given any vector X at a point m E M,

there exists a Lagrangian submanifold through the point tangent to X. We

6

1.

Symplectic Manifolds

shall see this as a corollary to the Darboux Theorem in the next section. Also Lagrangian submanifolds tend to get in the way of each other; loosely speaking two Lagrangian submanifolds that are Cl close tend to intersect more than one would expect of two arbitrary C' close submanifolds. Going into this point in some detail, let M be a compact manifold and, identifying M with the zero section of T*M, view M as a Lagrangian submanifold of T*M. Now let L be a Lagrangian submanifold of T*M near M. Regarding L as the image of a closed 1-form 0, the question of when L and M intersect reduces to the question of when 0 has a zero. So to perturb M to a disjoint Lagrangian submanifold L, M must admit a closed 1-form without zeros. An obstruction to this was given by Tischler [1970] in the following Theorem.

Theorem 1.1 If a compact manifold admits a closed 1 -form without zeros, then the manifold fibres over the circle and conversely.

In contrast the problem of perturbing M to an arbitrary disjoint submanifold is equivalent to finding a non-vanishing 1-form which is equivalent to finding a non-vanishing vector field and the obstruction to this is the Euler characteristic.

If M is simply connected, the situation is "worse". For now 0 is exact and hence given by a function on M, but since M is compact, such a function must have at least two critical points. So a perturbation of S2 to an arbitrary submanifold in T* S2 may intersect in only one point, but a perturbation to a Lagrangian submanifold must have at least two intersection points. For Ll and L2, C' close Lagrangian submanifolds of a symplectic manifold (M, S2), the same situation holds by virtue of the following theorem of Weinstein [1971] which we will prove in the next section.

Theorem (Weinstein) If L is a Lagrangian submanifold of a symplectic manifold (M, SZ), then there exists a neighborhood of L in M that is symplectomorphic to a neighborhood of the zero section in T*L.

More general than the notion of a Lagrangian submanifold are the notions of isotropic and coisotropic submanifolds. A submanifold t : N --f M2'n is isotropic if c*SZ = 0, so in particular the dimension of N is < n. In the subject of almost Hermitian manifolds (M2n, J, g) these submanifolds are called totally real submanifolds, Yau [1974], Chen and Ogiue [1974a]. The key point is that since Sl(X,Y) = g(X, JY), J maps the tangent space into the normal space. We remark that one sometimes sees another notion of totally real submanifold in the literature, namely a submanifold for which no tangent space contains a non-zero complex subspace; however we will use only the stronger notion in this text.

1.3 The Darboux-Weinstein theorems

7

The isotropic or totally real condition at a point m E N can be written as

a*TmN C_ {V IQ(V, c*TmN) = 0}. A submanifold c : N -p Men is coisotropic if c*TmN D {V IQ (V, c*TmN) = 0}; in terms of (J, g), J maps the normal space

into the tangent space, so the dimension of N is > n. In particular for a Lagrangian submanifold N'n in Cn, J maps the tangent spaces onto the normal spaces; therefore TCnIN = TNn ® iTNn = TN ® C and hence the complexified tangent bundle is trivial. Gromov [1971] (see also Weinstein [1977]) proved that if Nn is compact, then Nn admits a Lagrangian immersion into CCn if and only if the complexified tangent bundle is trivial. The question of imbeddings is a different matter. It is known that the sphere Sn cannot be imbedded in Cn as a Lagrangian submanifold. This is a consequence of a more general result of Gromov [1985] that a compact imbedded Lagrangian submanifold in Cn cannot be simply connected (see also Sikorav [1986]). For an immersed sphere as a Lagrangian submanifold with only one double point, see Example 5.3.3, Weinstein [1977, p. 26] or Morvan [1983].

Our discussion also has the following application to the problem of fixed points of symplectomorphisms, Weinstein [1977, p. 29]. Let (M, Q) be a com-

pact simply connected symplectic manifold. Then a symplectomorphsim f sufficiently C' close to the identity has at least two fixed points. To see this let A be the diagonal of (M, Sl) x (M, -Sl) and Ff the graph of f. Then A and If intersect at least twice and hence f has at least two fixed points. For example let M be complex projective space CPn and f an automorphism of the Kahler structure that is C' close to the identity. Any function on CPn has at least n + 1 critical points, so that f must have at least n + 1 fixed points. 1.3

The Darboux-Weinstein theorems

We have mentioned the theorems of Darboux and Weinstein already; in this section we present a modern proof of both theorems using a technique of Moser [1965]. The use of this idea to prove the classical Darboux Theorem is due to

Weinstein [1971, 1977] and independently to J. Martinet [1970]. In addition to the papers mentioned, a general reference is the book by P. Libermann and C.-M. Marle [1987, Clip. III, Section 15] which we follow here; for the Darboux Theorem see also N. Woodhouse [1980, pp. 7-9]. We begin with the following theorem of Weinstein. As a matter of notation, for a submanifold c : N --> M and a differential form 4) on M, 4) IN denotes the form acting on TNM, the restriction of TM to N, and not the pullback, t*4), of to M (see e.g., Libermann and Marle [1987, p. 360]). Theorem 1.2 Let Slo and SZ, be symplectic forms on a symplectic manifold M and N a submanifold (possibly a point) on which 1O(N = S21IN. Then there

1. Symplectic Manifolds

8

exist tubular neighborhoods U and V and a symplectomorphism p : U ---> V such that pl N is the identity. Proof. Since d(S21 - 520) = 0, by the Generalized Poincare Lemma (see e.g., Libermann and Marle [1987, p. 361]) there exists a tubular neighborhood W

of N and a 1-form a on W such that 521 - Do = da and aIN = 0. Now for t E R set Sgt = Qo + t(Sl1 - Slo). Sgt is non-degenerate on an open subset W1 of W x R containing N x R. Let X be the vector field on W1 defined by

X (M' t)J Qt (m) = -a(m) where J denotes the left interior product. For a point m, consider the integral curve t - 0m(t) of X through (m, 0) and regard the domain of cl : (m, t) --p

0n,,(t) as an open subset W2 of W x J with W2 C W1. Since aIN = 0, X restricted to N xR vanishes and hence N x R C W2. Now since [0, 1] is compact,

any point m e N has a neighborhood Ur,,, C M such that Urn x [0, 1] C W2. Let U = UmENUn,,. For (m, to) E U x [0, 1] we compute dt(clm(t)*Qt)(m)I t=to

= clm(to)*( X

to

dtQtl t=to)(m) = 0

since

£xQt = X-i dS2t + d(Xi Sgt) = -da = S2o - 521, and d 52t = 521 - S2o.

Thus if p : U -3 V = p(U) C W is the diffeomorphism determined by (p(m), 1)

then p*S21 = 52o and p(m) = m for m E N.

As a corollary we now have the classical theorem of Darboux.

Theorem 1.3 Given (M2n, S2) symplectic and m c M, there exist a neighborhood U of m and local coordinates (xl,..., xn, y1, ..., yn) on U such that

S2=TdxiAdyi. Proof. Let (u1, ..., un, v1, ..., vn) be local coordinates on a neighborhood of m such that au (m) and av; (m) form a symplectic frame at the point m. Set S20 = E2 and 01 = E dui A dvi. Then S2o and S21 agree at m. Now constructing

p as in the previous theorem, xi = ui o p and yi = vi o p form the desired coordinates.

N

We remark that one can easily choose the "Darboux" coordinates such that ayl (m) is any preassigned vector X at m and that xi = const., z = const. defines a Lagrangian submanifold. Thus we see that given a point m and a tangent vector X at m there exists a Lagrangian submanifold through the point and tangent to X, as we remarked in the last section. We now prove the theorem of Weinstein that locally a symplectic manifold is the cotangent bundle of a Lagrangian submanifold.

1.4 Symplectomorphisms

9

Theorem 1.4 If L is a Lagrangian submanifold of a symplectic manifold (M, Q), then there exists a neighborhood of L in M that is symplectomorphic to a neighborhood of the zero section in T*L.

Proof. Let TLM be the restriction of the tangent bundle TM to L and E a Lagrangian complement of TL in TLM. Such a vector bundle E exists but is by no means unique; e.g., relative to an associated metric as described above, E could be taken as the normal bundle of L. Define j : E -i T*L by

j(()(X) = c((,X) where c E Em and X E T,,,L, m E L. Moreover there exist a tubular neighborhood U of L in M and a diffeomorphism 0 of U onto ci(U) C E such that OIL is the zero section and, identifying Tq(m,)Em, with Em, 0*(m)IEm = idIEm,

MEL

(see e.g., Libermann and Marle [1987, p. 358] or in the Riemannian case use the inverse of the exponential map). Then j o 0 is a diffeomorphism of U onto

the open subset j(ql(U)) of T*L whose restriction to L is the zero section, so : L -> L' C T*L. Moreover (j o ci)*(m) maps the complementary Lagrangian subspaces Tm,L and Em onto Tso(.m)L' and Tso(.m)(TTL) respectively. But Tso(m)L' and Tso(m)(T, L) are complementary Lagrangian subspaces with

respect to the symplectic form d,6 on T*L. Now identifying L and L', the restriction of (j oci)*(m) to TmL is just the identity and since c*(m)IE,,, = idIE,,,,,

(j o ci)*(m) restricted to Em, is j. In particular (j o ci)*(m) is vertical and X E TmL, so using the local expression > dpi A dqi of d,6 on T*L,

o cb)*(m)(, (j o O)*(m)X) = j(()(X) _ Q((, X). The result now follows from Theorem 1.2.

1.4

Symplectomorphisms

Recall that a diffeomorphism f : M - M is a symplectomorphism if PQ = Q. A vector field X which generates a 1-parameter group of symplectomorphisms is called a symplectic vector field. Clearly £xIl = 0.

Theorem 1.5 Let X be a symplectic vector field on (M, SZ), g an associated metric and J the corresponding almost complex structure. Then Xi = Jikgk for some closed 1-form 9. Conversely, given a closed 1 -form 9, Xi = JikOk defines a symplectic vector field.

1. Symplectic Manifolds

10

2(Xi 0) is a closed 1-form. Let Proof. 0 = £XQ = d(XJ Sl) implies 0 = T be the vector field given by g(T, Y) = 0(Y). Then g(JT, Y) = -0(JY) _ -1l(X,JY) = g(X, Y). Therefore Xi = JikTk or Xi = JikOk as desired. Conversely given 9 closed, define X by Xi = Jik°k from which -01 = JiiXi = 2(Xi Q). Therefore £XIl = d(Xi St) = 2d9 = 0. -JiIXi, i.e., 0 =

Corollary 1.1 For f e C' (M), JV f is symplectic where V f is the gradient of f. Conversely, given X symplectic, X is locally JV f .

In particular X is locally the Hamiltonian vector field Xf. Compare this with the following classical treatment. Suppose that N is a level hypersurface of the function Hon (M, SI), on which dH 0. Then XH is a non-zero tangent vector field which is in the direction of J of the normal direction: g(Y, VH) _

YH = Q(XH, Y) = g(XH, JY) = -g(JXH, Y) giving XH = JVH. Finally we prove a result of Hatakeyama [1966] that the group of symplectomorphisms acts transitively on a compact symplectic manifold.

Theorem 1.6 The group of symplectomorphisms acts transitively on a compact symplectic manifold (M, S2).

Proof. We first prove the result for a Darboux neighborhood U about p E M, i.e., we have local coordinates (x', .., xn, yl, ..., y"l) such that SZ = > dxi A dyi

and xi(p) = yi(p) = 0. Let q(# p) E U with coordinates (ai, bi) and define E(aiyi - bixi). Then the vector field X defined a function f on U by f = 2 by X-i l = 2df generates a 1-parameter group Ot such that 01 (p) = q. For, writing X = Xi+ Xi*aye, Xi S1 = Xidyl - Xi*dxi = aidyi - bidxi. Thus X = ai a ;; + bi -58 and its integral curves have the form xi = ait, yi = bit. E(aiyi-bixi) Strictly speaking X is determined by f E C°°(M), where f = a on U and vanishes outside some larger neighborhood; M compact then implies that Ot is a diffeomorphism of M. Thus any two points in U may be joined by a symplectomorphism. Now for p, q E M join them by a curve and cover it by a finite number of Darboux neighborhoods Uq, a = 1, ..., k. Choose a sequence of points pa such that po = p,pk = q and pa E 14 n l.(,,+1 and apply the above

result.

For a generalization to symplectomorphisms mapping k points to k points, see Boothby [1969] or Kriegl-Michor [1997, p. 472].

2 Principal S1-bundles

2.1

The set of principal S'-bundles as a group

Let P and M be C°° manifolds, 7r : P -> M a C°° map of P onto M and G a Lie group acting on P to the right. Then (P, G, M) is called a principal G- bundle if

1. G acts freely on P, 2. 7r(pi) = 7r(p2) if and only if there exists g E G such that pig = P2, 3. P is locally trivial over M, i.e., for every m E M there exists a neighborhood U of m and a map Fu : 7r-'(U) -> G such that Fu(pg) = (Fu(p))g and such that the map : 7r-1(U) -p U x G taking p to (7r(p), Fu(p)) is a diffeomorphism.

For a general reference to the theory of principal fibre bundles see Bishop and Crittenden [1964, Chapters 3 and 5], Kobayashi and Nomizu [1963-69, Chapter II].

We now turn to the case where G = S', in which case we say that P is a principal circle bundle over M and we study the group structure of the set P(M, S1) of all principal circle bundles over M. Our treatment is based on Kobayashi [1956].

Given P, P' E P(M, S1) with projections 7,7r', let 0(P x P') = {(u, u') E P x P'17r(u) = 7'(u) 1. We say (u,, ui) ' (u2, u2) if there exists s E S1 such that u1s = u2 and uis-i = u2. Note that since S1 is abelian, u3 = u2t = ulst U3 = u2 t-1

= uis-1t-I = ui(st)-1.

12

2.

Principal S1-bundles

Let P + P' = 0 (P x P')/ -and 7r" : P + P' --> M the induced projection. S' acts on L(P x P') by (u, u') s = (us, u'). Now if (ul, u') N (u2i u2), u1t = u2

and u'1t-1 = u'2 and so u2s = silts = (uls)t. Therefore (uls, ui) - (u2s, u'2) and hence S1 acts on P + P'. S' acts freely: Suppose u"s = u", u" E P + P' and suppose (u, u') represents u". Then (u, u') - (us, u') so that u's-1 = u' and hence

s=1ES1. S1 acts transitively on fibres: Suppose ui, u2 E 7r"-1(m) and (ul, u'), (u2iu2) are representatives. Then u2 = U1 S, u2 = u'1s', s, s' E Si.

Now (u2i u2) - (u2s', ui) = (ulss', u') = (ul, ui)ss' and hence u"2 = u1ss " '.

P + P' is locally trivial: If Fu(u) = g, Fu' (u') = g', set Fu" (u, u') _

gg'. Then F/(us,u') = gsg' = gg's. Theorem 2.1 Under the operation +, P(M, S1) is an abelian group.

Proof. Let P0 be the trivial bundle and a : P -> P + Po defined by a(u) _ [(u, (7r(u),1))]. Then a is a bundle isomorphism:

a(us) = [(us, (7r(u),1))] = [(u, (7(u), 1))s] = [(u, (7r(u),1))]s = a(u)s; a-1([(u, (7F(u), g))]) = a-1([(ug-1, (n(u),1))]) = ug-1.

Let -P be a manifold diffeomorphic to P and -u the point corresponding

to u. Define -7r: -P --> M by -7r(-u) = 7r(u). S1 acts on -P by (-u)s = -(us-1). Then -P E P(M, Sl). Now let (ul, -u2) E 0(P x -P); then there exists a unique s E S1 such that ul = u2s. Let a : P+ (-P) -) Po be defined by a([(ul, -u2)]) _ (7r(ul), s). Then a is a bundle isomorphism. Let 0(P x P' x P") = {(u,u',u")I7r(u) = 7r'(u') = 7r"(0)} and define the equivalence - by (u, u', u") - (us, u's-1 s', u"s'-1). Then 0(P x P' x P")/ is naturally isomorphic to (P + P') + P", ((u's-1, us)s', u"s'-1), and to P + (P' + P"), (us, (u's', a"s'-1)s-1). S1 acts on 0(P x P' x P") by (u, u', u")s = (us, u' u"). Now if (u1,u' u") '- (U2,2 u' 2u") then u2 = 1u t u' = u't-1t' u"2 = 1 1 2 u'1ti-1. Then u2s =silts = (u1s)t so that the right action preserves Finally P + P' is isomorphic to P' + P by [(u, u')] F-----> [(u', u)], (us, u') 1

(u, u's).

'.

U

Let G, be the cyclic subgroup of S1 of order m and P E P(M, S1). Since S1 acts on P on the right, so does Gm. Then P/C,, is a principal bundle over M with group S1/Gm,. But S1/G,,,, = S1 and hence we can consider PIG, E P(M, S'). More precisely: Let [u] be an element of P/Gm which is

2.1 The set of principal S'-bundles as a group

13

represented by u E P. Define the action of S1 on P/Gm by setting [u] s = [us'] where s = s''n. This definition is independent of the choice of u and s'. For

if g E Gm, [ug] s = [ugs'] = [us'g] = [us'] = [u] s and if s""° = s, then (s'-ls" )tm = 1 so that s'-ls" E G,,.,, and hence [us"] = [us's'-1s"] _ [us'].

Theorem 2.2 Let P, Gm and P/Gm be as above. Then P/Gm m P. Proof. From the definition above it follows by induction that m P can be defined directly by A(Px...xP)={(ul,...,u,,,,)EPx...xPl7r(ul)=...=7r(u,,,L)I

two elements of which, say (u1, ..., u,,,,) and (uls1,..., umsm) are equivalent if

and only if Si s,,,, = 1. The quotient space of 0(P x . x P) by this relation is m P. The action of Sl on m P is given by [(ul, ..., u,,,,)] s = [(uls, u2i ..., um)]. Define 0 : P/Gm --) m P by ql([u]) = [(u, ..., u)] which is independent of the choice of u, for if g E Gm, gm = 1, then 0([ug]) = [(ug,..., ug)] = [(u, ..., u)] Now take s E S1 and s' such that s'm = s, then .

.

O([u]s) = O([us']) = [(us', ..., us')] = [(us, u, .., u)] = [(u, ..., u)]s = Therefore 0 is a bundle isomorphism of P/Gm onto m P

(0([u]))s.

Corollary 2.1 If P is simply connected and m > 1, then there is no bundle

P' E P(M, S1) such that P = m P. Proof. Suppose that P' exists, then P ^_' P'/Gm and so P' is a covering space of P. Since P is simply connected, this can happen only if m = 1. A principal bundle may also be thought of as an equivalence class of principal coordinate bundles which are given by their transition functions. Let {U2} be a differentiably simple open cover of M (i.e., {U } is locally finite, each u2 has compact closure and any nonempty finite intersection is diffeomorphic

to an open cell of Rn). With respect to this cover let fij : u2 n Uj -p S' be the transition functions of a bundle P E P(M, Sl). The fij are defined by fij (7 (p)) = Fu (p) (Fu, (p))-1. Then fij E F(U2 n U S'), the set of all sections over u2 n Uj with coefficients in S1,the sheaf of germs of local C00 maps

from M into S1. Thus f = { ffj} is a cochain of M. Now ffk = f2jfjk. Thus fap2122 = 5f2021 = f2021fio 2fi122 = f2022f,p 2 = 1 and f is a cocycle. Now P and

P' are equivalent if and only if P - P' is the trivial bundle, so P and P' are the same here if and only if f fi-1 is a coboundary where f is the cocycle of P'. Therefore P(M, S')' H1(M, Si).

The natural short exact sequence, 0 -* Z --> R -> Sl 0 induces a short exact sequence of the corresponding sheaves 0 -f 2 -p 7Z -# Sl --> 0. From this we get the cohomology sequence

... - H1(M,R) - H1(M,S1) - H2(M,Z) - H2(M,R)

...

.

2. Principal S1-bundles

14

Now let {Oi} be a partition of unity subordinate to {Ui} with ciIM_vi = 0, Vi C Ui. Let {,Qijk} E Z2 (M' R.). Consider aij = >k Okl3ijk- If in a neighborhood of m c Ui n Uj, /3ijk is not defined, we have qlk(m') = 0 for every m' in the neighborhood. Hence we see that aij is defined on Ui n 14. Now

CS{aij} _ {aij - aik + ajk} = {L 01(Aj1 - Akl + Njkl)} OlIijk} = {Qijk}.

This can be done for other-dimensional cocycles as well. Hence Zi(M, R) _ Bi (M, R), i > 0. Thus in particular we have that H1 (M, R) = H2(M, R) = 0. On the other hand the C°° maps of M into Z are just the constant integer functions and hence the corresponding cohomology groups must be isomorphic, in particular H2 (M, Z) = H2 (M, Z). Thus we finally have the isomorphisms

H'(M, S') ^' H2(M, Z),

P(M, SI) = H2(M,Z).

For example it is well known that for complex projective space CPn, H2 (CPn, Z) ^_' Z. Thus by the above isomorphism, P (CPn, Sl) = Z. The most famous example of a principal circle bundle over CPn is the Hopf fibration of an odd-dimensional sphere S2n+1. Suppose that S2n+1 E P(CPn, Sl) corresponds to k E Z, then since Stn+1 is simply connected, k = ±1 by the Corollary. Therefore we finally have the following result.

Theorem 2.3 P(CPn, S') = Z with S2n+1 corresponding to 1 for a proper orientation of CPn.

2.2

Connections on a principal bundle

A connection on a principal G-bundle (P, G, M) is a C°° distribution H on P such that

1. TpP=Hp6Vp, Vp=ker7r*, 2. Rg*(Hp) = HP9 where R9 is right translation. Vectors in Hp are said to be horizontal and for t E TpP we denote its horizontal part by Ht. The map 7r*I HP is one-to-one and hence 7r,k(Hp) = TT(p)M. Thus

given a vector field X on M there exists a unique vector field k on P such

2.2 Connections on a principal bundle

15

that X(p) E Hp and 7rX(p) = X(7r(p)), i.e., X is or-related to X. X is called the horizontal lift of X. The following two properties of horizontal lifts follow easily from the definitions:

[X, Y] = H[X,Y],

Rg*X = X.

A p-form w on P is vertical (resp. horizontal) if w(ti, ..., tp) = 0 when one or more of the ti's is horizontal (resp. vertical). Now regard p E P as a map of G - ) P by p(g) = pg. Let g denote the Lie algebra of G and define a Lie algebra homomorphism of g into a Lie algebra g of (p*X)(e). p : G -p ir-1(ir(p)) is a diffeomorphism, vector fields on P by

so given t E V, let X (e) = p*'t; then there exists X E g such that X (p) = t. On the other hand, given X E g, 7r*X (p) = ir* (p*X) (e) = (7r o p)*X (e) = 0, since 7r o p is a constant map. Thus X is vertical.

Given a connection H on (P, G, M) define a g-valued 1-form 0 on P by q(t) = X E g where 9(p) is the vertical part of t. 0 is called the connection form of H. The following lemmas are well known and their proofs can be found in the references (e.g., Bishop and Crittenden [1964, pp. 76-77], Kobayashi and Nomizu [1963-69, p. 64] )

Lemma 2.1 0 o Rg* = Ad g-1 o 0, i.e. 0 is equivariant. Lemma 2.2 If qs is a g-valued C°° equivariant 1-form such that 0(9(p)) = X, then there exists a unique connection H whose 1 -form is 0. Given a g-valued p-form o- and a g-valued q-form w, their bracket is defined by [9,w](X1,...,Xp+q)

1

(p + q)!

L sgn(ii, ..., ip+q) [O'(Xi1, ..., Xip), w(Xip}1, ..., Xip+q)]' (i1,...,2P+q)

Let w be a p-form on P and define a (p + 1)-form Dw by Dw(tl,..., tp+1) = dw(Ht1, ..., Htp).

Clearly Dw is horizontal. If 0 is the connection form of H, I = Do is called the curvature form of H.4) is equivariant as can be seen as follows. 4'(R9*t1, R9*t2) = dcb(HRg*ti, HR9*t2) = d0(Rg*Htl, R9*Ht2)

= 12 O(Rg*[Hti,Ht2]) = -1Adg-10([Htl,Ht2]) 2

Adg-11)(t1,t2)

We now have the structural equation; again for the proof see the references (e.g., Bishop and Crittenden [1964, p. 81], Kobayashi and Nomizu [1963-69, Chapter II, p. 77]).

16

2.

Principal SI-bundles

Theorem 2.4 do = -; [0, 0] +,D. Let P E P(M, S') and note that the Lie algebra of Sl is R with the trivial bracket operation. Thus if rl is a connection form on P, [rl, rl] = 0 and so if is the curvature, the structural equation is simply drl = (D. Again since Sl is abelian, for X, Y E T,1P and s E Sl we have ' (Rs*X, Rs*Y) = Ads-1(D(X, Y) = (D (X, Y).

Therefore there exists a unique 2-form Il on M such that 1 _ it 1. Now 7r*(dS2) = dD = 0 and hence Q is a closed 2-form M. If now rl' is another connection, then as before (r) - rl') o Rs* = rl - if.

Hence there exists a unique 1-form ,Q on M such that 7r*/3 = rl - if. Now 7r*d,13 = d(rl - r;') = 1D - V = 7r*52 - 7r*52' and hence d O = S2 - S2'. Thus the cohomology class of S2 is independent of the choice of the connection form and

is called the characteristic class of P; again see Kobayashi [1956]. Since the transition functions are mappings from Uz n U, into S', they can be considered as real-valued functions (mod 1) and it can then be shown that Q is integral, i.e., ff S2 = integer for any finite singular cocycle c with integer coefficients. This gives a homomorphism of P(M, S') onto the integral classes of the second deRham cohomology. We end this chapter with the following theorem of Kobayashi [1963] which

should again be compared with the isomorphism P(M, S1) = H2(M, Z). Theorem 2.5 Let Il be a 2 -form on M representing an element of H2(M, Z); then there exist a principal circle bundle P and a connection form 'q on P such that dry = it*f2.

Proof. Let P be the principal bundle corresponding to Sl and if a connection form on P so that the closed 2-form S2' defined by dry' = 7r*S2' is cohomologous to I. Let ,Q be a 1-form on M such that S2 - S2' = d,3. Now set rl = r)' + rr*J3.

7* is horizontal and equivariant and hence rl is a connection form on P and drl = lr*S2 as desired.

S

3 Contact Manifolds

3.1

Definitions

By a contact manifold we mean a C°° manifold Men+1 together with a 1-form 0 is a volume element 0. In particular 77 A (drl)n rl such that 77 A (drl)n

on M so that a contact manifold is orientable. Also drl has rank 2n on the Grassmann algebra AT LM at each point m E M and thus we have a 10 and 0}, on which rl dimensional subspace, {X E which is complementary to the subspace on which q = 0. Therefore choosing 1 we have a global vector field Sm in this subspace normalized by satisfying &7(e, X) = 0,

rl

1.

is called the characteristic vector field or Reeb vector field (Reeb [1952]) of the contact structure rl. Computing Lie derivatives by the formula £ = d o (ei) + (e ) o d we have immediately that £E77 = 0,

cdrl = 0.

We denote by D the contact distribution or subbundle defined by the subspaces Dm = {X E TmM : rl(X) = 0}. Roughly speaking the meaning of the contact condition, rl A (drl)n 0, is that the contact subbundle is as far from being integrable as possible; in particular D rotates as one moves around on the manifold. For a subbundle defined by a 1-form rl to be integrable it is necessary and sufficient that rl A (drl) - 0. In contrast we shall see in Chapter 5 that, for a contact manifold Men+1, the maximum dimension of an integral submanifold of D is only n. A one-dimensional integral submanifold of D will

3. Contact Manifolds

18

be called a Legendre curve, especially to avoid confusion with an integral curve

of the vector field . A contact structure is regular if is regular as a vector field, that is, every point of the manifold has a neighborhood such that any integral curve of the vector field passing through the neighborhood passes through only once (cf. Palais [1957]). Two well-known examples of non-regular vector fields on surfaces are the irrational flow on a torus and the flow around a Mobius band. We now prove the classical theorem of Darboux.

Theorem 3.1 About each point of a contact manifold (M2n+1, r)) there exist xn yl local coordinates (xl yn, z) with respect to which n

= dz -

yidxi.

Proof. In some coordinate neighborhood choose a 2n-ball transverse to ; d77 is symplectic on this ball and hence there exist local coordinates (x', ... , xn, y1, ... , yn, u) such that dry = E dxi A dyi. Now d(r) + E yidxi) = 0 so that ri + E yidxi = df for some function f, i.e., y = df - E yidxi. Now 77 A (dr7)n =

0. Therefore df is independent of

df n dx1 A

. A dxn n dy1 A

dx',...

dyl,... dyn and hence we can regard x', yi and z = f as a

, dxn,

. A dyn

,

coordinate system.

In a Darboux coordinate system = az For if = a 0 + bi a ti + ci ay, , then a - biyi. Also 0 = dxi A 1= a ) gives ci = 0. Similarly 0 = drl(', ay;) gives bi = 0. Thus a = 1 and so = aZ A diffeomorphism f of Men+1 or between open subsets of R2n+1 with the contact structure of the Darboux form of Theorem 3.1 is called a contact .

transformation if f *77 = T77 for some non-vanishing function T on the domain

of f. If T - 1, f is called a strict contact transformation. There is also the notion of a contact structure in the wider sense, often called simply a "contact structure" by many authors, which can be defined in a number of ways. For example, a contact manifold in the wider sense is a manifold with a differentiable structure modeled on the pseudogroup of contact

transformations on R2+1 (J. Gray [1959] ), i.e., in the overlap of coordinate neighborhoods the transition functions preserve the Darboux form to within a non-vanishing function multiple. An alternate approach is to put the emphasis on the field of 2n-planes D and to define the structure as a hyperplane field defined locally by a contact form. In the overlap of coordinate neighborhoods U n 1.f', r7' = f *77 = T17 and hence dr7' = f *d77 = dT A rl + Tdr7 from which / 77

A (d77')" = Tn+17 A (d77)n 0 0.

3.1 Definitions

19

Let M2n+1 be a contact manifold in the wider sense. On a coordinate neighborhood Lfa choose coordinates (x1, ... , x2n+1) such that rla A (d77a)n = A dx2n+1 with Aa > 0. Similarly on a neighborhood Up choose aadx' A coordinates (y1, .. y2n+1) such that q,3 A (d9p)n = )pdyl A . A dy2n+1 with )gyp > 0. Now on 74, f1 Llp rla = Tpr)p and hence 7l A (dr)a)n = T' 171p A (d7jp)n. -

Therefore Teal \,31 axe I = Aa. From this one may easily obtain the following results (see also J. Gray [1959], Sasaki [1965], Stong [1974]).

Theorem 3.2 Let M2n+1 be a contact manifold in the wider sense. If n is odd, then M2n+1 is orientable.

Theorem 3.3 Let M2n+1 be a contact manifold in the wider sense. If n is even and M2n+1 is orientable, then M2n+1 is a contact manifold. We also have the following theorem of Sasaki [1965].

Theorem 3.4 Let M2n+1 be a contact manifold in the wider sense which is not a contact manifold in the restricted sense; then its 2-fold covering manifold is a contact manifold in the restricted sense.

Proof. Let {Lfa} be an open cover of M2n+1 by coordinate charts. Recall the local contact forms rla above and their transition 77a = Taprjp. Consider the M2n+1 given as follows. M2,+1 is the union of 2-fold covering 7r : M2n+l the sets {U,,, X Z2} with the following equivalence relation, where Z2 is the set of integers ±1. Let ea denote +1 or -1. Two elements (pa, Ea) E U", X Z2 and (pp, Ep) E Up X Z2 are equivalent if pa = pp and ea = sgn(Tap(pp))Ep. Now define local contact forms on ua x ±1 by rl(a Eo) = Ea7r*rla. Then ea7r*77a = Ea7r*(Tap7p) = Ec(Tap O 7r)7r*7),3 = Eaep(Tap 0 7r)7l(p,Ep)

and eaep(Tap o 7r) > 0. The local forms 71(a,E,) can now be used to construct a global contact form on M2n+1

In particular a connected and simply connected contact manifold in the wider sense is a contact manifold in the restricted sense; for an alternate approach to this idea see Monna [1983]. The name contact (Beruhrungstransformation) seems to be due to Sophus Lie [1890] and is natural in view of the simple example of Huygens' principle (Huygens [1690]; see also MacLane [1968, part II, p. 83]). Consider R2 with

coordinates (x, y). The classical notion of a "line element" of R2 is a point together with a non-vertical line through the point. Thus a line element may be regarded as a point in ]183 determined by the point and the slope p of the line. Given a smooth curve C in the plane without vertical tangents, say y = f (x), its tangent lines determine a curve in R3 with coordinates (x, y, p)

20

3. Contact Manifolds

which is a Legendre curve of the contact form y = dy - pdx. If now C is a wave front, by Huygens' principle the new wave front Ct at time t is the envelope of the circular waves centered at all the points of C, say of radius t taking the velocity of propagation to be 1. Corresponding to a point (x, y) on C, the point (x, p) on Ct lies on both the normal line and the circle of radius t centered at (x, y), i.e., p - y = -1(x - x) and (x - x)2 + (y - y)2 = t2. Thus (x - x)2 = P2+1, so depending on the direction of propagation, e.g., choosing the negative root, the transformation of R3 mapping (x, y, p) to

x=x-

pt

p=y+

p'-

t

+1

p2'

p=p,

maps C to Ct. A simple calculation shows that dy - pdx = dy - pdx and so the transformation is a contact transformation. Moreover tangent wave fronts

(curves) are mapped to tangent wave fronts (curves) and hence the name "contact".

3.2 Examples In [1971] J. Martinet proved that every compact orientable 3-manifold carries a contact structure. However, we now have the following theorem of J. Gonzalo [1987] showing that there are three independent contact structures. Theorem 3.5 Every closed orientable 3-manifold has a parallelization by three contact forms. Before turning to some detailed examples we should mention that, in contrast to Martinet's result, there exist (2n+1)-dimensional manifolds, n > 2, with no contact structure even in the wider sense. In particular, Stong [1974] showed that for every n > 2 there is a closed oriented connected manifold of dimension

2n + 1 with no contact structure in the wider sense. One such manifold is SU(3)/SO(3) for n = 2 and (SU(3)/SO(3)) X S2n-4 for n > 2. 3.2.1

R2n+1

xn Y11. In effect we have already seen that R2n+1(x1 yn, z) with the Darboux form rl = dz - E? 1 yidxi is a contact manifold. The characteristic vector field is 0Z and the contact subbundle D is spanned by I

Xi

i = 1,...

, n.

a

3

= axi +y'az,

a Xn+i = ayi,

3.2 Examples

21

Rn+1 x PRn

S. 2.2

We now give an example of a contact manifold in the wider sense which is not a contact manifold in our sense (J. Gray [1959]). Consider Rn+1 with coordinates (xi, ... , xn+1) and real projective space PRn with homogeneous coordinates,

to+1) and let M2n+1 = ]E8n+1 x PRn. The subsets U, i = 1,...,n+ 1

(t1,

defined by ti

0, form an open cover of M2n+1 by coordinate neighborhoods. 1 n+1

On Ui define a 1-form 77i by rii =

ti j=1

tjdx'; we then have i n (thi)n 54 0 and

77i. Thus, M2n+1 has a contact structure in the wider sense, but for n

77i = "2

even, M2n+1 is non-orientable and hence cannot carry a global contact form. 3.2.3 M2n+1 C R 2n+2 with TmM2n+1 n{o} = 0

Turning to more standard examples, we prove the following theorem (J. Gray [1959]).

Theorem 3.6 Let t

: M2n+1

]R 2n+2 be

a smooth hypersurface immersed in R2n+2 and suppose that no tangent space of M2n+1 contains the origin of IR2n+2' then M2n+1 has a contact structure.

Proof. Let xA, A = 1,... , 2n + 2 be cartesian coordinates on R2n+2 and consider the 1-form (k = x1dx2 - x2dx1

Let V1, (xo

+ ... + x2n+1dx2n+2 _ x2n+2dx2n+1

... V2n+1 be 2n + 1 linearly independent vectors at a point xo = ,

x2on+2) and define a vector W at x0 with components

WA = *dxA(V1' ... , V2n+1) R2n+2. Then where * is the Hodge star operator of the Euclidean metric on W is normal to the hyperplane spanned by V1, ... , V2n+1. Now regard x0 as a vector with components xoA. then

(an(dc)n)(V1,... ,V2n+1) _

xAWA. 0

Thus if no tangent space of M2n+1 regarded as a hyperplane in R2n+2 contains the origin, then 77 = C'c is a contact form on M2n+1.

As a special case we see that an odd-dimensional sphere S2n+1 carries a contact structure. Moreover ci is invariant under reflection through the origin, (xi, ... ,x2n+2) -* (-X11... , -x2n+2) and hence the real projective space

22

3. Contact Manifolds

PR2"+1 is also a contact manifold. J. A. Wolf [1968] then considered more general quotients of S2n+1 and proved that a complete connected odd-dimensional

Riemannian manifold of positive constant curvature inherits a contact structure from the form a. xidxn+l+i and denote by Rn+1 Similarly consider the 1-form a and 1R2+1 the subspaces defined by xi = 0 and xn+l+i = 0 respectively, i = 1, ... , n + 1. Then ,Q induces a contact form on Men+1 if and only if Men+1 n R1+1 = 0 and Men+1 n I1g2+1 is an n-dimensional submanifold and no tangent space of Men+1 n R2+1 in R2 +1 contains the origin of R2 " .

More generally, given a symplectic manifold (M, SI), a hypersurface c : S -M is said to be of contact type if there exists a contact form 77 on S such that dry = c*S2.

3.2.4

T1 M, T1M

We shall show that the cotangent sphere bundle and the tangent sphere bundle of a Riemannian manifold are contact manifolds (see e.g., Reeb [1952], Sasaki [1962]). Let M be an n + 1-dimensional Riemannian manifold and T*M its cotangent bundle . Also let (x', ... , xn+1) be local coordinates on a neighborhood U of M and (pl, ... , pn+l) coordinates on the fibres over U. If it : T*M --> M is the projection map, then as in Chapter 1, qi = xi o it and pi are local coordinates on T*M. Consider the Liouville form 3 which is given locally by ,Q = En1 pidgi. The bundle T, *M of unit cotangent vectors has empty intersection with the zero section of T*M, its intersection with any fibre of T*M is an n-dimensional sphere and no tangent space to this intersection contains the origin of the fibre. Thus, as in the discussion at the end of the last example, ,Q induces a contact structure on the hypersurface T, *M of T*M.

Similarly one obtains a contact structure on the bundle T1M of unit tangent vectors. In fact if Gi, denotes the components of the metric on M with respect to the coordinates (x', ... , xn+1) and if (v1, ... , vn+1) are the fibre coordinates on TM, define 0 locally by ,Q = Ei J Gi;vi dqi where qi = xi o7r and 7: TM M is the projection (see Section 9.2 for details).

-f

3.2.5 T*M X R

Let M be an n-dimensional manifold and T*M its cotangent bundle. As in the previous example we can define a 1-form Q by the local expression Y 1 pidgi. Let Men+1 = T*M x R, t the coordinate on R and y : Men+1 T*M the projection to the first factor. Then q = dt - u*,Q is a contact form on Men+1

3.2 Examples

23

3.2.6 T3

We have mentioned that Martinet proved that every compact orientable 3manifold carries a contact structure. Here we will give explicitly a contact structure on the 3-dimensional torus V. First consider R3 with the contact form 77 = sin ydx + cos ydz;

i7 A dry = -dx A dy A dz.

= sin y ax + cos y az and D is spanned by Iay , cos y ax - sin y-& }. The rotation

of D in the direction of the y-axis is dramatically clear in this example. Thus one sees the non-integrability of D as one moves around on the manifold even though D does not rotate when one moves along the line of intersection of D with a plane y = const. Now y is invariant under translation by 27r in each coordinate and hence the 3-dimensional torus also carries this structure. For each value of y, e induces a flow on the 2-torus defined by y = const. Depending on the value of y the flow is a rational or irrational flow on T2. Thus the contact structure on T3 is not regular. We will see in Theorem 4.14, that no torus carries a regular contact structure. Note here in particular though, that some of the integral curves of 6 are closed and some are not.

3.2.7 T5 In [1979] R. Lutz proved the existence of contact structures on principal T2bundles over 3-manifolds. In particular the 5-dimensional torus admits a contact structure given by the form rl = (sin 01 cos 93 - sin 92 sin 93)d84 + (sin 01 sin 03 + sin 92 cos 03)d05 + sin 92 cos 92d01 - sin 01 cos 91d92 + cos 01 cos 02d93.

Hadjar [1998] obtains other explicit contact forms on the 5-torus, especially ones for which the contact subbundle is transverse to the trivial fibration of T5 over T4 by circles.

There has been also been recent interest in constructing 5-dimensional contact manifolds by using other surfaces as fibre. Both Geiges [1997a] and Altschuler and Wu [2000] have proved the following theorem using very different techniques.

Theorem 3.7 Let M be a compact orientable 3-manifold and E a compact orientable surface. Then M x E carries a contact form. Altschuler and Wu also show the existence of contact forms on the products

S2p+k+3 X Mk x E where Mk is compact, orientable and parallelizable and E

a compact orientable surface as before.

24

3. Contact Manifolds

3.2.8

Overtwisted contact structures

Examples 3.2.1 and 3.2.6 on R3, namely q _ (dz - ydx) and i _ (sin ydx + cosydz), have cylindrical coordinate versions (see also Douady [1982/83, p. 131], Bennequin [1983, p. 93]). Let (r, 0, z) be the usual cylindrical coordinates

on R3 \ {x = y = 0}. Making the naive substitutions y -; r,

dx - rd0,

dz -f dz these examples become 17 A dry = -2rdr A d9 A dz

zt = dz - r2dG, 19

D

(9z'

a, a

{ ar

aB

2

az

and

1= cos rdz + r sin rd8,

17 A dry _ (r + sin r cos r)dr A d9 Adz,

a rcosr+sinr a sin r r + sin r cos r aB + r + sin r cots r az' D={

ar, cosre- rsinrz

Note that in both examples the integral curves of a are Legendre curves

and in the second example that the curve r = ir, z = constant is also a Legendre curve. Thus in the second example D is tangent to the disk A = {z = 0, r < 7r} C R3 along the boundary. Now consider the topological disk OE = {z = Ere, r < 7r}. D is tangent to AE only at the origin. On DE\ {(0, 0, 0)}

define a line field by the intersection of the tangent plane to the paraboloid DE and V at each point. These fields can be expressed by the vector fields

in the first case and

in the second. For simplicity take the projection of these vector fields to the xyplane. The integral curves in the first case are the logarithmic spirals r = AeTE and in the second case are the curves 9 = -2E In sin r+C which near the origin spiral indefinitely and approach a limit cycle on the boundary of the disk. Thus we have the following diagrams; a diagram of this type for the second example was introduced by Douady [1982/83] (see also Bennequin [1983, p. 94]) and we will call such a diagram the Douady portrait of the contact structure.

3.2 Examples

25

A 3-dimensional contact manifold is said to be overtwisted (Eliashberg [1989]) if there exist a contact embedding of a neighborhood of the disk A with the contact structure 77 = cos rdz + r sin rdO. Roughly speaking the meaning of overtwisted is that as one moves radially from the image of origin, the plane D turns over in a finite distance. For the

radial Legendre curves in the disk A, D turns over as one goes from r = 0 to r = 7r. This means that for some r between 0 and 7r the vector field is tangent to the disk A, and in particular for r the solution of r cos r + sin r = 0 in (0, 7r), r 2.02876, the circle of this radius in A is an integral curve of . 3.2.9 Contact circles

At the beginning of this Section we mentioned the result of Gonzalo [1987] that a compact orientable 3-manifold has three independent contact structures. Geiges and Gonzalo [1995] introduce the notion of a contact circle: A 3-manifold admits a contact circle if it admits a pair of contact forms (771, 772) such that for any (Al, A2) E S1, A1771 + A2772 is also a contact form. This circle is a taut contact circle if the contact forms A1771 +A2772 define the same volume form for all (A1, A2) E S'; this is equivalent to the two conditions 771 A d771 = 772 A d772,

771 A d772 = -772 A d771.

Geiges and Gonzalo then prove the following classification theorem.

Theorem 3.8 A compact orientable 3-manifold admits a taut contact circle if and only if it is diffeomorphic to the quotient of a Lie group G by a discrete subgroup, acting by left multiplication, where G is either SU(2), the universal cover of PSL(2, R) or the universal cover of the group of Euclidean motions E(2).

In contrast to the overtwisted contact structures in Example 3.2.8 (and in contrast to taut contact circles), a contact structure is said to be tight if it is not overtwisted (see e.g. Eliashberg and Thurston [1998]). Geiges and Gonzalo [1995] also prove that the connected sum of any number of copies of

26

3. Contact Manifolds

the manifolds listed in Theorem 3.8, T2-bundles over Si or S2 x S1 admit a contact circle consisting of tight contact structures.

3.3

The Boothby-Wang fibration

We now give an important class of examples, namely principal circle bundles over symplectic manifolds of integral class and conversely we will prove the celebrated theorem of Boothby and Wang [1958] that a compact regular contact manifold is of this type. Examples of this type are often simply referred to as Boothby- Wang fibrations. In Chapter 2 we saw that the set of principal circle bundles over a manifold M has a group structure isomorphic to the cohomology H2(M,Z). Now let (M2n, Q) be a symplectic manifold such that [S2] E H2(M2n,Z) and 7r M2n+1 3 M2n the corresponding circle bundle. By Theorem 2.5 there exists a connection form rl on M2n+l such that drl = 7r*S2. Now if is a vertical vec1, and X1, ... X2n linearly independent horizontal tor field, say with vector fields, then (ri A (drl)n) X1,. .. , X27') is non-zero. Thus regarding the Lie algebra valued form rl as a real-valued form, we see that rl is a contact structure on M2n+1 The most well-known special case of the Boothby-Wang fibration is the Hopf fibration of an odd dimensional unit sphere S2n+1 over complex projective space CPn of constant holomorphic curvature equal to 4. The standard contact structure on S2n+1 obtained in Example 3.2.3 can also be obtained by the above construction. Additional details of the geometry of Boothby-Wang fibrations will be given from time to time, particularly in Examples 4.5.4 and :

,

6.7.2.

Theorem 3.9 Let (M2,,+1 g') be a compact regular contact manifold; then there exists a contact form rl = Trl' for some non-vanishing function Tr whose characteristic vector field generates a free effective S1 on M2n+1 Moreover M2n+1 is the bundle space of a principal circle bundle 7r : M2n+1 -p M2n over a symplectic manifold M2n whose symplectic form S/ determines an integral cocycle on M2n and i is a connection form on the bundle with curvature form drl = 1r*Q.

Proof. Since q' is regular, its characteristic vector field ' is a regular vector field and hence its maximal integral curves or orbits are closed subsets of M2n+1; but M2n+1 is compact, so these integral curves are homeomorphic to circles. Moreover, as l;' is regular, M2n+1 is a fibre bundle over a manifold M2n (the set of maximal integral curves with the quotient topology, see e.g., Palais [1957]) and we denote the projection by 7r.

3.3 The Boothby-Wang fibration

27

M2n+1 denote the 1-parameter group of diffeomorphisms generated by ' and define the period A' of E' at m E Men+1 by A'(m) = inf{tit > 0, ft(m) = m}. A' is constant on each orbit and since there are no fixed points, A' is never zero. Also as the orbits are circles, A' is not infinite. We will show that A' is constant on all of M2n+1. Our argument is due to Tanno [1965]. Let k be a Riemannian metric on M2n and let g = 7r*k-f-7710 r7'. Then g is a Riemannian metric on Men+1 and E' is a unit Killing vector field with respect to g since = 1 and £g r7' = 0. If V denotes the Levi-Civita 0 and hence the orbits of are connection of g, g(Ve't;', X) = geodesics. If -y is an orbit through m, let y' be an orbit sufficiently near to y that there exists a unique minimal geodesic from m to y' meeting 'y' orthogonally at m'. Then since ft is an isometry for all t, the image of the geodesic

Now let ft

:

Men+1

arc mm' is orthogonal to y and ' for all t. Thus, as a point m on y moves through one period along y, the corresponding point on y' moves through one period and hence A' is constant on Men+1 Now define 77 and by 77 = a, rT and _ A' is constant, e is the characteristic vector field of the contact form r/. Moreover has the same orbits as ' and its period function A = 1. Thus the one-parameter group ft of depends only on the equivalence class mod 1 of t and the action so induced of Si is effective and free. Since is regular we may cover Men+1 by coordinate neighborhoods with coordinates (z1, ... , x2n+1) such that the integral curves of are given by

xl = const.,... , x2n = const. Projecting such neighborhoods we obtain an open cover {Ui} of Men, and on each Ui we define a local cross section si by setting x2n+1 = const. Then define Fi : Ui x S' -; S1 by Fi(p, t) = si(p)t. The transition functions for the bundle structure are then given by fi; (p) _ Fi(p, t)F,(p, t)-1. We have already seen that £gr7 = 0 and 0, so that rl and dr7 are invariant under the action of S1. Now take A = dt as a basis of 651 = R, the Lie algebra of S1 and set r7 = 77A so that q may be regarded as a Lie algebravalued 1-form. For an element B E 61 denote by B* the induced vector field on

Men+1. In particular A* = , so that r"7(A*) = A. Moreover right translation

by t E S1 is just ft so that Rtr7 = r7 by the invariance of 77 under the S' action. Thus, g (precisely ) is a connection form on M2n+1

If S2 is the curvature form of 77, then the structural equation is d77 = - 2 [r7, ri] + 1 = S2 since S' is abelian. On the other hand dr7 is horizontal and invariant, so there exists a 2-form S2 on Men such that dr7 = 7r*Q. Now 7r*dS2 = drr*S2 = d2r7 = 0 so that dSZ = 0 and rr*(S2)n = (7r*5 )n = (dr7)n 0 giving Qn ; 0. Therefore Men is symplectic. Finally, since the transition functions fib are real(mod 1)-valued, [S2] E H2(M2n,Z) (see e.g., Kobayashi [1956]).

28

3.4

3. Contact Manifolds

The Weinstein conjecture

In Example 3.2.6 we studied a contact structure on the 3-dimensional torus and observed that some of the orbits of are closed and some are not. It is a wellknown conjecture of Weinstein [1979] that on a compact contact manifold M satisfying Hl (M, IR) = 0, l; must have a closed orbit. The present author knows of no example of a compact contact manifold not satisfying H' (M, R) = 0 for which e does not have a closed orbit and believes the conjecture is true without

the assumption of H' (M, R) = 0. In view of the example on the torus the following result of Petkov and Popov [1995] is interesting. Let (M2n+1, i) be an analytic, connected contact manifold with complete characteristic vector field . Since 97 n (dri)T is invariant under the action of 1;, it induces an invariant Lebesgue measure on Men+1. A point m E M2ri+1 is a periodic point if the integral curve of through m is periodic. Petkov and Popov prove that either

the set of periodic points has measure 0 or there exists T > 0 such that (m) = m for every point m. Note that for some m, T could be a multiple of the period for that m (cf. the notion of an almost regular contact structure below). Interest in the Weinstein conjecture has often been phrased in terms of the question of the existence of periodic orbits of Hamiltonian systems. In Section 1.1 we considered briefly a real-valued function H on a symplectic manifold (M, S2) and its Hamiltonian vector field XH. Since XHH = 0, XH is tangent to the level (energy) surfaces of H. Thus if a level surface S is of contact type, the Hamiltonian vector field XH is collinear with the characteristic vector field e.

For hypersurfaces in R2n+2 with the standard symplectic structure, the Weinstein conjecture is known to be true if the hypersurface is convex (Weinstein [1978]), star-shaped (Rabinowitz [1978]) and, more recently, if the hypersurface is of contact type and without the assumption Hl (M, ll8) = 0 (Viterbo [1987], see also Hofer and Zehnder [1987]). As an aside it is interesting to note that Ginzburg [1995] showed that if one gives up on the hypersurface being of contact type, then there exist imbeddings of S2,,,+1, n > 3, in R2n+2 whose Hamiltonian vector field has no closed orbits.

In cotangent bundles T*M, first note that if S is a compact connected hypersurface, then T*M \ S has exactly two components, one of which is bounded; Hofer and Viterbo [1988] prove that if the bounded component of T*M \ S contains the zero section and if S is of contact type, then 1; has a closed orbit.

Turning to compact contact manifolds in general (not necessarily hypersurfaces of contact type), Hofer [1993] showed that the Weinstein conjecture

3.4 The Weinstein conjecture

29

is true on a compact overtwisted contact manifold, on the 3-sphere for any contact form and on a closed orientable 3-manifold with ir2

0.

Banyaga [1990] showed that the Weinstein conjecture is true if the contact form is C2 close to a regular contact form (again without assuming H1(M, R) = 0). C. B. Thomas [1976] introduced the notion of an almost regular contact structure. A contact structure is said to be almost regular if there exists a positive integer N such that every point has a neighborhood such that any integral curve of passing through the neighborhood, passes through at most N times. With this idea in mind Banyaga and Rukimbira [1994] showed that the Weinstein conjecture is true if the contact form is C' close to an almost regular contact form. A contact manifold is called an R-contact manifold (Rukimbira [1993]) if is Killing with respect to some (not necessarily associated (Chapter 4)) Riemannian metric g for which i(X) = g(X, ). For compact contact manifolds admitting such a metric the Weinstein conjecture is true (Rukimbira [1993]). An auxiliary result of Rukimbira [1994] is that for a compact K-contact manifold Men+1 (see Subsection 4.5.4 or Section 6.2), the dimension of any leaf closure is at most the smaller of n + 1 and 2n + 1 minus the rank of the vector space of harmonic vector fields. Also on a compact K-contact manifold, has at least n + 1 closed orbits (Rukimbira [1995a]) and a compact simply connected K-contact Men+1 with exactly n + 1 closed orbits is homeomorphic to a sphere (Rukimbira [1999]).

4 Associated Metrics

4.1

Almost complex and almost contact structures

We will generally regard the theory of almost Hermitian structures as well known and give here only definitions and a few properties that will be important for our study; many of these were already mentioned in Chapter 1. For more detail the reader is referred to Gray and Hervella [1980], KobayashiNomizu [1963-69, Chapter IX] and Kobayashi-Wu [1983]; also, despite its classical nature, the book of Yano [1965] contains helpful information on many of these structures.

An almost complex structure is a tensor field J of type (1,1) such that J2 = -I. A Hermitian metric on an almost complex manifold (M, J) is a Riemannian metric which is invariant by J, i.e.,

g(JX, JY) = g(X,Y) Noting that J is negative-self-adjoint with respect to g, i.e., g(X, JY) _ -g(JX, Y), S2(X, Y) = g(X, JY) defines a 2-form called the fundamental 2form of the almost Hermitian structure (M, J, g). If M is a complex manifold and J the corresponding almost complex structure, we say that (M, J, g) is a Hermitian manifold. If df = 0, the structure is almost Kdhler. For geometers working strictly over the complex domain, a Hermitian metric is a Hermitian quadratic form and hence complex-valued; it takes its non-zero values as appropriate when one argument is holomorphic and the other anti-holomorphic. Our metric g becomes half the real part of g(X-iJX, Y+iJY). For geometers concerned with a variety of structures presented for the most part in terms of real tensor fields, as we will be, it has become quite standard to use the word

32

4. Associated Metrics

Hermitian as we have done. A reader interested in this point may want to see Kobayashi-Wu [1983, pp. 80-81] for commentary. Note also that every almost complex manifold admits a Hermitian metric, for if k is any Riemannian metric, g defined by

g(X, Y) = k(X, Y) + k(JX, JY) is Hermitian.

Given (M, J, g) we can construct a particular local orthonormal basis as follows. Let U be a coordinate neighborhood on M and X1 any unit vector field on U. Let X1. = JX1. Now choose a unit vector field X2 orthogonal to both X1 and X1.. Then JX2 is also orthogonal to X1 and X1 . Continuing in this manner we have a local orthonormal basis of the form {X1, ... , Xn, JX1 7... JXn}. Such a basis is called a J-basis. Note in partic,

ular that an almost complex manifold is even-dimensional. Again given (M, J) choose g Hermitian. Let {Ua} be an open cover with J-bases {Xz, Xi. }, {Xj, X. } on U, and Up respectively. With respect to these bases J is given by 0

(I

I

0

If now X E TmM, m E U. n Up, then for the column vectors of components (X) and (X) with respect to these bases

(X)=(C where A, B, C, D are n x n matrices and (

(I

0

)(C

)(x) C

D ) E 0(2n). Now

)(x)=(x)=( C

D)(JX)

A B

(C D/ CI i.e., ( C

BD ) and

and hence (

C D)

(0

-)

0

)(X)

commute. Therefore D = A and C = -B

E U(n). In particular the structural group of the tan-

gent bundle of an almost complex manifold is reducible to U(n). Recall also

that det

( B B)

manifold is orientable.

_ det(A + iB) I2 > 0 and therefore an almost complex

4.1 Almost complex and almost contact structures

33

Conversely suppose that we are given M such that the structural group of TM can be reduced to U(n). Let {U,,} be an open cover such that we can choose local orthonormal bases which transform in the overlaps of neigh-

borhoods by the action of U(n). In each {U,,} define J. by (

j p ); this

matrix commutes with U(n) and hence the set {J} determines a global tensor field J such that j2 = -I. Thus an almost complex (Hermitian) structure on M can be thought of as a reduction of the structural group to U(n). As we will see in our discussion of associated metrics below, the structural

group of a symplectic manifold is reducible to U(n) and that of a contact manifold to U(n) x 1 (Chern [1953]). For an odd-dimensional manifold Men+1 J. Gray [1959] defined an almost contact structure as a reduction of the struc-

tural group to U(n) x 1. In terms of structure tensors we say Men+1 has an almost contact structure or sometimes (0,1;,,q)-structure if it admits a tensor field 0 of type (1,1), a vector field t; and a 1-form rl satisfying

02=-I+77

77

(e)=1.

Many authors include also that cal; = 0 and rl o

= 0; however these are

deducible from the other conditions as we now show.

Theorem 4.1 Suppose Men+1 has a rl o 0 = 0. Moreover the endomorphism

l;, a7)-structure. Then has rank 2n.

= 0 and

Proof. First note that 02 =

0 and rl(t;) = 1 give 02t oof and hence either cal; = 0 or fit; is a non-trivial eigenvector 0 corresponding

to the eigenvalue 0. Using 02 = -I + 77 ® again we have 0 = 02g _ -06 + +77(06)e or 06 = 77(06)6. Now if 06 is a non-trivial eigenvector of the eigenvalue 0, q(06) 0 and therefore 0 = 026 = 77(06)06 = (77(06))2e 0, a contradiction. Thus, 456 = 0.

Now since 06 = 0, we also have that rl(gX)l; = 03X + OX = -OX + 0(77(X)6) + OX = 0 for any vector field X and hence rl o 0 = 0. Finally since 06 = 0, 6 0 0 everywhere, rankq5 < 2n + 1. If a vector field 6 satisfies O = 0, 02 = -I + 77 ®6 gives 0 = - + 77(66; thus l; is collinear with 6 and so rankq5 = 2n. If a manifold Men+1 with a (0, 6, 71)-structure admits a Riemannian metric g such that g(OX, OY) = g(X, Y) - 77(X)77(Y),

we say Men+1 has an almost contact metric structure and g is called a compatible metric. Setting Y = 1; we have immediately that 71(X) = g(X, ).

34

4. Associated Metrics

As in the case of an almost complex structure the existence of the compatible

metric is easy. For if k' is any metric, first set k(X, Y) = k'(q52X, O2Y) + 77(X)77(Y); then i7(X) = g(X,t;). Now define g by g(X, Y) = 2 (k(X, Y) + k(OX, OY) + 77(X)77(Y))

and check the details. For a manifold Men+1 with an almost contact metric structure (q, 17, g) we can also construct a useful local orthonormal basis. Let U be a coordinate neighborhood on M and X1 any unit vector field on U orthogonal to l;. Then X1. _ OX1 is a unit vector field orthogonal to both X1 and . Now choose a unit vector field X2 orthogonal to 1;, X1 and X1.. Then qX2 is also a unit vector field orthogonal to , X1, Xi. and X2. Proceeding in this way we obtain a local orthonormal basis {Xi, Xi. = OXi, } called a 0-basis. Now given a manifold Men+1 with a (¢, , 77)-structure, let g be a compatible metric and {14} an open cover with O-bases {Xi, Xi., }. With respect to such a basis 0 is given by the matrix 0 -I 0

1I

0

0

0

0

0

Proceeding as in the almost complex case, we see that the structural group of Men+1 is reducible to U(n) x 1. Conversely given an almost contact structure as defined by this reduction of the structural group and an open cover {U,} respecting the action of U(n) x 1, define 0,, on ua by the above matrix and define 77, and l;a by row and column vectors of zeros with last entry 1. Then, again as in the almost complex case, this defines global structure tensors (0, , 77) satisfying 02 = -I +,q ® and 1. We shall subsequently speak of an almost contact structure r7) and suppress the terminology "(0, , r7)-structure".

4.2

Polarization and associated metrics

We begin with a discussion of the well-known decomposition, called "polarization", of a non-singular matrix A into the product of an orthogonal matrix

F and a positive definite symmetric matrix G. Let H(n) denote the set of positive definite symmetric n x n matrices and as usual O(n)the orthogonal group. In treating the subject of constructing Riemannian metrics associated to 2-forms of rank 2r, Y. Hatakeyama [1962] proved the analyticity of the polar decomposition; that the decomposition is continuous can be found in

4.2 Polarization and associated metrics

35

Chevalley [1946, p. 14-16]. We prove Hatakeyama's result by a sequence of Lemmas.

Lemma 4.1 For G E H(n), the map o-G

:

g[(n, R) -+ g((n, R) given by

QG(A) = GAG-1 has positive eigenvalues.

Proof. Let Ai > 0 be the eigenvalues of G. There exists P E O(n) such that PGP-1 is diagonal, say A. Then ap-1o"po-P = o-G and hence oo and o-G have the same eigenvalues. Now o (A)il = (DAD-1)il = Ejk Aisijajkskl A = ai ail Thus the n2 eigenvalues of o-G are the positive numbers i .

Lemma 4.2 For G E H(n) and A skew-symmetric, AG symmetric implies

that A=0. Proof. AG = GT AT = -GA. Therefore CG (A) = -A and so by the previous lemma A = 0.

Now O(n) and H(n) are analytic submanifolds of GL(n, R); thus cp : O(n) x H(n) --> GL(n, R) defined by co(F, G) = FG is analytic.

Lemma 4.3 dco is one-to-one and hence cp-1 given by polarization is analytic.

Proof. Let X E T(F,G)O(n) x H(n) and consider the curve (FetA, G + tB) where A is skew-symmetric and B is symmetric and which has tangent X at (F, G).

dcp(X) = tim

FetA(G + tB) - FG

= FAG + FB.

t If now dco(X) = 0, F(AG + B) = 0. Therefore AG = -B which is symmetric. Thus by Lemma 4.2, A = 0 and hence also B = 0.

Theorem 4.2 Polarization as a map from GL(n, R) - O(n) x H(n) gives an analytic diffeomorphism between these manifolds with respect to the usual analytic structures. We now prove the existence of associated metrics.

Theorem 4.3 Let (M2n, S2) be a symplectic manifold. Then there exists a Riemannian metric g and an almost complex structure J such that g(X, JY) = Q(X, Y).

Proof. Let k be any Riemannian metric on M and let {X1,... X2, } be a local k-orthonormal basis. Let Aij = 1(Xi, Xj). A is a 2n x 2n non-singular ,

skew-symmetric matrix. By polarization we have A = FG for some orthogonal matrix F and positive definite symmetric matrix G. Now define g and J by g(XX, Xj) = Gij,

JXi = F'Xj.

36

4. Associated Metrics

g is independent of the choice of k-orthonormal basis. For if { Y , ,

.

.

. ,

Yen } is

another k-orthonormal basis, there is an orthogonal matrix P such that B1 = Sl(Y,Y;) = S2(PkiXk,P' X,) = PkiPljAkl = (PAP-1)2,j. If B = (DI is the polarization of B, (DI' = PFP-'PGP-1 and so by the uniqueness of the polar decomposition 1) = PFP-1 and I' = PGP-1. Thus, in particular, we see that g and J are globally defined. Also since A is skew-symmetric,

F2 = -I and F is skew-symmetric. To see this note that AT = GFT = -FG and hence applying F on the right, G = -FFFT GF. But FT GF is positive definite symmetric and so the uniqueness of the decomposition gives -F2 = I.

Finally F = -F-' = -FT

.

In particular given a symplectic manifold (M2", Q) we say a Riemannian metric g is an associated metric if there exists an almost complex structure J such that g(X, JY) = Sl(X,Y). We remark that if g is an associated metric and one uses it as the starting metric k in the above polarization process, the process yields the metric g back again. Theorem 4.4 Let (M2n+1 77) be a contact manifold and its characteristic vector field. Then there exists an almost contact metric structure such that g(X, cY) = drl(X, Y).

Proof. This time the proof is a two step process. First let k' be any Riemannian metric and define a new metric k by

k(X, Y) = k'(-X +

-Y +

77(X)i7(Y).

Then k(X,e) = r7(X). Now polarize dr7 on the contact subbundle D as in the symplectic case. This gives a metric g' and almost complex structure 0' on D such that g'(X, ql'Y) = d77 (X, Y). Extending g' to a metric g agreeing with k in the direction and extending 0' to a field of endomorphisms 0 by requiring O = 0, we have an almost contact metric structure (qi, , 77, g) such that g(X, OY) = d97(X, Y).

As in the symplectic case, given a contact manifold (M2-+I, 77) we say a Riemannian metric g is an associated metric if there exists an almost contact metric structure such that g(X, qY) = d?7(X, Y). In this case we also speak of a contact metric structure; other authors often use the phrase contact Riemannian structure. Working strictly with structure tensors, one may avoid the polarization process, by defining an associated metric as follows: Given a contact manifold (M2n+1,q) with characteristic vector field , a Riemannian metric g is an associated metric if there exists a tensor field of type (1,1), such that 02 = -I + 17 0 ,

77(X) = g(X, ),

d77(X, Y) = g(X, OY).

4.2 Polarization and associated metrics

37

Finally we caution that it is possible to have a contact manifold (M2n+1, y) with characteristic vector field and an almost contact metric structure (0, , rl, g), same and q, without g(X, cY) = drl(X, Y); see Example 4.5.3 for an example. In the course of this book we will give many properties of associated metrics; we give one simple property here as it is discussed in Example 4.5.5 and used periodically.

Theorem 4.5 On a contact metric manifold the integral curves of

are

geodesics.

Proof. For a contact metric structure we have

0 = (So) (X) = g(X, ) - g(V X -

g(X, Vet)

so the integral curves of are geodesics.

In our discussion above in both the symplectic and contact cases we started with an arbitrary Riemannian metric and obtained an associated metric. Thus we are led to believe that there are many associated metrics to a given sym-

plectic or contact form. Indeed this is the case and we now show that the set A of all associated metrics is infinite dimensional by exhibiting a path of metrics in A determined by a C°° function with compact support. Such paths of associated metrics will be useful to us in the study of critical points of curvature functionals on A. We give the construction in the symplectic case and then remark on the similarity with the contact case. Let f be a C°° function with compact support contained in a neighborhood U of (M2n, Q) and {X1, ... X., X,. , ... X,.,. l a local J-basis. Let g be an associated metric. Make no change in g outside U and on U change g only in the planes spanned by {X1, X1. } by the matrix

(

1+tf+2t2f2 1t2f2

2t2f2

1-tf+2t2f2

This defines a path of metrics gt and it is easy to check that each gt is an associated metric for the symplectic form Q. In the contact case simply begin with a 0-basis and make the same construction. Also, as already remarked in Section 1.1, it is evident that in the symplectic case, A may be thought of as the set of all almost Kahler metrics that have S2 as their fundamental 2-form.

On the other hand it is possible for a Riemannian metric g to be an associated metric for more than one symplectic structure. For example on a hyper-Kahler manifold one has, by definition, three independent global Kahler structures (Jd, g), a = 1, 2, 3, satisfying J1J2 + J2J1 = 0, J3 = J1J2. Thus the

38

4. Associated Metrics

three fundamental 2-forms give three symplectic structures with g an associated metric for each of them. We close this section by showing that all associated metrics have the same volume element and give the proof only in the symplectic case. Theorem 4.6 Let (M2n, Q) be a symplectic manifold (resp. (M"'+', 77) a contact manifold) and g an associated metric. Then

dV =

(2nn)InQn,

(resp. dV =

(2

n'n77

n (d77)n))

Proof. Let X1,... Xn, X1.... , Xn. be a J-basis and 01,...

,

the dual basis. Then with respect to this dual basis dV = 91 A

on, 91`, 01*

n

n

... A0"A9n*

...

9n*

A 02 A 92` A

9'n9'*. Thus i=1

i=

Sln = (-2)n((9' A 91`) + ... + (9n A 9n'))n

_ (-2)nn!(91 A 91.) A ... A (9n A 9n`) _ (-2)nn!dV.

4.3

Polarization of metrics as a projection

In the previous section we created associated metrics from an arbitrary metric by polarization. In this section we give some further properties of the set A and discuss how A sits in the set Ar of all Riemannian metrics with the same volume element. Restricting ourselves to V we will interpret the polarization process of constructing associated metrics from a given metric as a projection from M onto A.

On a compact manifold M the set M of all Riemannian metrics may be given a Riemannian metric (Ebin [1970]): The tangent space, T9.M, of M at a metric g is the space of symmetric tensor fields of type (0,2). For symmetric tensor fields S and T of type (0,2) we define a Riemannian metric (.,.) by

(S, T)9 = I SijTklg'kgjldVg. M

For the set M of metrics with the same volume element, the geodesics in Al were found by Ebin [1970] and are curves of the form gent where S is symmetric

with tr S = 0. gt = gest is computed by

gt(X,Y) = g(X,estY)

4.3 Polarization of metrics as a projection

39

where here est acts on Y as a tensor field of type (1,1). Again and throughout we shall often denote a symmetric tensor field of type (0,2) and the corresponding tensor field of type (1,1) by the same letter. In fact in most of our computations we begin with a local orthonormal basis for some g E A and regard S simply as its matrix of components. As an aside, we remark that one can think of metrics with the same total volume on an n-dimensional manifold as being at a fixed distance from the zero tensor. Consider the path tg, t E [0, 1] from the zero tensor to the metric g; then since g may be considered as the tangent to the path at each point, we have the following entertaining computation.

IgI = J (g, g) g2dt = f 1 (2 4.3.1

to det g dxl ... dxn)1/2dt = fAM

Some linear algebra

For the projection result below, Theorem 4.8, we will need the polarization of

a particular path in GL(2n, R). Let J denote the matrix (0

- ) and let

S be any symmetric 2n x 2n matrix. First diagonalize S, say Q-1SQ = A, Q E O(2n) A diagonal, and set P = Qe-2et. Our problem will be to polarize PT,7P, i.e., find F(t) E O(2n), G(t) E H(2n) such that PT,7P = F(t)G(t).

Lemma 4.4 Any symmetric 2n x 2n matrix S can be uniquely written as B + D, where B and D are symmetric and ,7B - B,7 = 0 and JD + DJ = 0.

Proof. Setting B = 2(-,75,7+S) and D = 2(JSJ+S) the decomposition is immediate. If now B + D = B' + D', B - B' = D' - D and hence ,7(B B') = (B - B'),7 gives 7(D'- D) = (D' - D),7 = -,7(D'- D). Therefore ,7(D' - D) = 0 and so, since J is non-singular, D' = D and in turn B' = B. Of course B and D need not commute with each other. We will see below that if [B, D] = 0, then F(t) = Q-1,7Q and G(t) = Q-1e-BtQ and conversely if either F(t) or G(t) have this simple form, B and D commute. As a matter of notation, using the analyticity of the polar decomposition, we write

F(t) = E F(k)tk, k=0

G(k) tk.

G(t) = k=0

At t = 0, PT,7P = Q-1JQ which we denote by M. Since J E O(2n), F(0) = M and G(0) = I. The main lemma of this section is the following; due to its complexity we refer to the author's paper [1983] for its proof.

40

4.

Associated Metrics

Lemma 4.5 F(k)

=2

k-1 ((_i)k (k)

-F(k-j)G(j)

Q-1j[(B - D)k-j, (B + D)']Q

-

MG(j)F(k-j)M

MF(k-j)F(j))

+

k-1

G(k)

= MF(k) + M E F(k-j)G(j) j=1

+(k!2_)k

Q-1(E

(j')(B-D)k-i(B+D)i)Q.

From this lemma we see easily that F(1)

= 0,

F(2) = 4Q-1,7[B,D]Q.

Continuing we can find F(k) and G(k) as far as desired; in particular G(1)

G(3)

= -Q-'BQ,

G(2)

= 1Q-1B2Q,

= Q-1(-1B3- 1 (BD2-2DBD+D2B)-1(B2D-2BDB+DB2))Q' 6

G(4)

24

8

1 (BD - B2DB - BDB2 + DB3)

= Q-1(24B4 +

+96 (2B2D2 - 7BDBD + 3DB2D + 7BD2B

- 7DBDB + 2D2B2)) Q.

Corollary 4.1 [B, D] = 0 implies F(t) = Q-1,7Q and G(t) =

Q-1e-BtQ

Conversely either of these implies the commutativity.

Proof. [B, D] = 0 implies that the bracket in F(k) vanishes and hence by induction F(k) = 0 for k > 0. Again if B and D commute k

()(B - D)k-' (B + D)'

= 2kBk

Q-1e-BtQ and hence G(t) = Clearly F(t) = Q-1 JQ gives F(2) = 4Q-1 J[B, D]Q = 0 and hence [B, D] 0. Finally if G(t) = Q-1 eBtQ ,

QG(3)Q-1 + 61 B3 = 0

QG(4)Q-1 - 24B4 = 0.

_

4.3 Polarization of metrics as a projection

41

Thus if we multiply the rest of the expression for GO) on the left by 48B and separately on the right by 48B and add these two to 96 times the rest of G(4) we have

-3BDBD + 3DB2D + 3BD2B - 3DBDB = 0. Thus [B, D]2 = 0, but [B, D] is skew-symmetric and hence [B, D] = 0.

Lemma 4.6 Jes = e-SJ if and only if SJ + JS = 0. Proof. The sufficiency is clear, so we prove only the necessity. Suppose SX =

AX, then JeSX = e"JX = e-SJX. Also let {ei} be an orthonormal eigenvector basis of S with Sei = niei. If now JX = Yiei, e-SJX = e'JX = e"Yxei and

e-SJX =

e_SYzei

=

Thus for each i, either \ = -ai or Yi = 0 and hence

SJX =

Y'Aiei = -AYzei = -AJX,

but JSX = AJX giving (SJ + JS)X = 0 for any eigenvector X and hence

4.3.2

Results on the set A

First we will give a remark about general curves in A. If gt = g + V11t + D(2)t2 + , g E A is a path of metrics, we can easily obtain a sequence of necessary conditions for gt to lie in A. We adopt the convention that D will signify a tensor field of type (0,2) or type (1,1) related by the metric, it being clear from context which is meant. We give the details in the symplectic case; the reader can easily read 0 instead of J for the contact case and follow

the computation, showing also that D(k) = 0 by a small amount of further computation. 00

g(X, JY) = Q(X, Y) = gt(X, JtY) = g(X, JtY) + E D(k) (X, JY)tk k=1

from which

J = Jt+D(1)Jtt+D(2)Jtt2+ Applying Jt on the right and J on the left we have

Jt = J(I + D(1)t + D(2)t2 +

42

4. Associated Metrics

Squaring this and comparing coefficients gives k-1

JD(k) + D(k) J =

- E D(j) JD(k-j) j=1

Using this last result repeatedly we see that

JD (k) + D(k) J = J(polynomial in the D(')'s, j < k). In particular we have JD(1) + D(1)J = 0,

JD (2) +D (2) j = JD (1)2.

These equations yield immediately the following corollary.

Corollary 4.2 A contains no line of metrics. Recall that in the contact case the construction of associated metrics was a two-step process, the first being the creation of a metric k such that k(X, Z;) _

,q(X) from an arbitrary metric k' and the second the polarization of drl restricted to the contact subbundle. The first step leads to a linear subspace G of

M. If ko and k1 are two Riemannian metrics for which ki(X, ) = rl(X), i =

0, 1, then k = (1 - t)ko + tk1 for all t for which k is positive definite is also a Riemannian metric with this property. Moreover if ko and k'1 yield ko and kl respectively under the first step of the process, (1 - t)ko + tk'1 yields

k= (1-t)ko+tk1. We have also seen that all associated metrics have the same volume element. On a symplectic manifold of dimension 2, A = N and on a contact manifold of dimension 3, A = N n L. In higher dimensions A is a proper subset of N (N n G). We shall now show that A is totally geodesic in N and then study how polarization of a path kt E N acts as a projection onto gt e A.

Theorem 4.7 A is totally geodesic in N (N n G) in the sense that if g E A and D is a symmetric tensor field satisfying DJ+ JD = 0 (Do+OD = 0 and D = 0), then gt = geDt lies in A.

Proof. Let Jt = JeDt (ot = OeDt) and note that JeDt = e DtJ. Then gt(X, JtY) = g(X, eDtJeDtY) = g(X, JY) = SZ(X, y) (= drl (X, Y)) and

Jt = JeDtJeDt = J2 = -I (0i = -I + rl ® ). In particular the tangent space, T9A, of A at g is the set of all symmetric tensor fields that anti-commute with J (0 and annihilate ).

4.3 Polarization of metrics as a projection

43

We now turn to the problem of understanding how the polarization of a geodesic in N gives rise to path gt E A and hence of viewing polarization as a projection of N onto A (the author [1983]). We suppress mentioning the contact case and for that case all tensors T should be understood as also satisfying T = 0.

Let us start with a geodesic kt = gent in JV where g E A and S any symmetric tensor field of vanishing trace. As we have seen if S and J anticommute, kt is already the path gt in A. We shall see as a corollary that if S and J commute, kt collapses to the metric g. (One can think of a matrix

B which commutes with J = C I j

)

I

as being orthogonal to a matrix D

which anti-commutes with T by showing tr BD = 0.) Writing S as B + D we will see that the construction process takes kt to geDt if and only if B and D commute, but that gt and geDt agree through second order in general. Let {Xi} be/a local g-orthonormal basis with respect to which J is given by the matrix (

I

I

.

The first problem is to construct a kt-orthonormal

basis. As remarked above we also denote by S the matrix of S with respect to

the initial basis {Xi}. If SX = \X, estX = e'tX and hence the eigenvalues of est are analytic in t. Let Q be an orthogonal matrix diagonalizing S as in Subsection 4.3.1, i.e., Q-1SQ = A, and set A = e-2"t and P = QA so that pTeStp = I. Let Xi(t) = Pki(t)Xk, then kt(Xi(t), Xj(t)) = g(PkiXk, estPljXi) =

PkiPlj(est)-lgk,,,

= PTestP = I.

Thus {Xi(t)} is a kt-orthonormal basis, but note that Xi(0) = QkiXk. Now our job is to polarize A(t) = cl(Xi(t), Xj (t)) = PT JP, giving F(t) and G(t) as in Lemma 4.5.

If we are to have an expression for gt that we can compare more easily for each t, we should express g(t) with respect to the original basis {Xi}. Now G(t) = gt(Xi(t), Xj(t)) = PkiP1igt(Xk, X1) = PT (gt(Xk, Xl))P, but as P = QA we have =QA-1G(t)A-1Q-1. gt(Xk,Xl) Theorem 4.8 If kt = gent is a geodesic in Al through g E A, then the path in A obtained by polarization is given with respect gt = g+D(1)t+D(2)t2+ to the basis {Xi} by J

D(`) = j 1 (g) (B + ! 1 kk j=0

k=0

G('-j) being given by Lemma 4.5.

D)7-kQG(l-j)Q-1(B + D) k'

4. Associated Metrics

44

The proof is by expansion of gt(Xk, X1) = QA-'G(t)A-'Q-1 using the series expansions of A-' = e2°` and G(t) and noting that A = Q-1SQ (again see the author [1983]).

We remark that Q need not be unique in the above argument (e.g., if S does not have distinct eigenvalues), but each G(k) is of the form Q-1 (polynomial in B and D) Q

and therefore D(k) is independent of the orthogonal matrix Q diagonalizing S. Again we list the first few terms: D(1)

=D

D(2)

D2

=

D(3)

2 '

=

D3 6

- 112(B2D - 2BDB + DB2),

D(4) = D+ 1 (-4B2D2+5BDBD+3BD2B-5DB2D+5DBDB-4D2B2). 24

96

Corollary 4.3 [B, D] = 0 if and only if gt =

geDt.

Proof. If [B, D] = 0, then as in Corollary 4.1 QG(k-.5)Q-1 =

(-1)k-j Bk-9

(k-j)! Thus =Ei(B+D)'(kl)

j)IBk-i

D(k)

j=o

j=o

Conversely if D(k) = 1,Dk, multiply the rest of the expression for D(3) on

both the left and right by -48D and add these two to 96 times the rest of D(4) to yield -3BDBD + 3DB2D + 3BD2B - 3DBDB = 0 and in turn the conclusion as in Corollary 4.1.

Corollary 4.4 If kt = gent and S commutes with J, gt = g. Proof. In this case D = 0 and the result follows from the previous corollary.

Theorem 4.9 Two metrics in A may be joined by a unique geodesic.

Proof. Let go and g1 be two metrics in A. From what has been said so far, the problem is to find S such that g1 = goes and SJo + JoS = 0. As before let

4.4 Action of symplectic and contact transformations

45

{Xi} be a local go-orthonormal basis with respect to which JO is given by the

matrix j= I

0

1 J. Then

.7 = Q(Xi, X,) = g1(Xi, J1X,)

from which g1 = -,7J1 where here J1 and g1 are regarded as the matrices of J1 and g1 with respect to the basis {X2}. In particular -,7J1 is positive definite symmetric and hence there exists a unique real symmetric matrix S

satisfying es = -JJ1. Then gl = goes and, since Ji = -I, ,7eS,7eS = -I from which e SJ = Jes. Lemma 4.6 then gives S,7 + ,7S = 0.

4.4

Action of symplectic and contact transformations

We begin by showing that if f is a symplectomorphism or strict contact transformation and g an associated metric, then f *g is also an associated metric.

Theorem 4.10 Let (M, Q) be a symplectic manifold, or respectively (M, 77) a contact manifold and f a diffeomorphism satisfying f *Q = S2, respectively f *r7 = rl. Then for any associated metric g, f *g is also an associated metric.

Proof. In the symplectic case define J* by (f *g) (X, J*Y) = 0(X, Y). Then

g(f*X, f*J*Y) = S2(X, Y) = Sl(f*X, f*Y) = g(f*X, Jf*Y)

and therefore f*J* = Jf*. Now f*J*2X = J2 f*X = - f*X, so that J* is an almost complex structure satisfying (f *g) (X, J*Y) = S2(X, Y) and hence f *g is an associated metric.

In the case of a strict contact transformation, first note that =71(x) = 1,

dil(e,x) = 0

and therefore f* = . Now define 0* by (f *g) (X, O*Y) = drl(X, Y) and proceed as in the symplectic case to get f*ql* _ Of* and in turn 0*2 = -I + r7

Also it is easy to check that (f *g) (C X) = 77(X ).

There is a partial converse for the case where M is a compact symplectic manifold and the diffeomorphism belongs to the connected component of the identity of the diffeomorphism group; in general the converse is not true and we will also give a couple of counterexamples (cf. Apostolov and Draghici [1999]). The diffeomorphism group of a compact manifold will be denoted by Di f f and the connected component of the identity by Di f fo. The group of symplectomorphisms will be denoted by S.

46

4. Associated Metrics

Theorem 4.11 Let (M, 52) be a compact symplectic manifold and g an associated metric. If for a diffeomorphism f E Di f fo, f *g is also an associated metric, then f E S. Proof. For the associated metrics g and f *g let J and j be the corresponding almost complex structures. Thus (J, g, S2) and (J, f *g, 52) are almost Kdhler structures and consider the action of f -1 on the second structure, i.e., setting J* = f*Jf -1*, (J*, g, f -1*Q) is again an almost Kahler structure with the same metric g. Since the fundamental 2-form of a compact almost Kahler manifold is harmonic, both Sl and f -1*SZ are harmonic with respect to the metric g. Now f E Di f fo and so 5l and f -1*Sl represent the same cohomology class in H2(M, R). Therefore Sl - f -1*SZ is exact, but since it is also harmonic, it must be zero by the Hodge decomposition. Thus f -1*5l = Sl giving f e S. In general the above theorem is not true and we give two counterexamples. First consider the diffeomorphism f of R4 given by x' = 1(x1 + x2 + x3 + x4), x2 = 2(-xl+x2-x3+x4x3 = 2(x1+x2-x3-x4), x4 =

2c-x1+x2+x3-x4

and the symplectic form Sl = 2(dx' A d;r- 3 + dx2 A dx4). Let g be the standard Euclidean metric on Il84 which is clearly an associated metric. Now it is easy to see that C axti

l

c 80(4) but not in Sp(4 R). Thus f *g = g but f *5l

S2;

in fact PQ = -S2. For a compact counterexample consider almost Kahler structures of the form (M = M1 X M2, g = 91 + 92, SZ = Q1 + 02) where (Ml, g1, 521) is any almost Kahler manifold and (M2 = S' X S1, 92, 522) is the standard product

Kahler structure on S1 x S1. Let f be the diffeomorphism idyl, x 0, where 0 is the map on S1 x S1 that interchanges the two factors. Clearly, f is an isometry of g, but it is not a ±-symplectomorphism, as f *52 = SZ1 - 522. In [1970] Ebin proved a "slice theorem" for the set of Riemannian metrics

M, i.e., given a metric g E M there is a neighborhood of g in A4 that is the product of a neighborhood of g in the orbit of g under the action of the diffeomorphism group and a submanifold orthogonal to the orbit with respect to the inner product on M. The tangent space to this submanifold or `slice' at g is the kernel of the codifferential S of g acting on second order symmetric

tensor fields (Berger-Ebin [1969]). In Theorem 4.10 we saw that if f E S and g E A, then f *g E A and we may consider the quotient space A/S. Considering the group of isometries that are also symplectic transformations, Smolentsev [1995] shows that the slice theorem of Ebin can be restricted to give a slice theorem for A.

This is a good point to give a technical lemma for use in Chapter 10, to show the Berger-Ebin result that a symmetric tensor field D orthogonal to an orbit of Di f f is in the kernel of the codifferential and to show at least that

4.4 Action of symplectic and contact transformations

47

a symmetric tensor field D is orthogonal to an orbit of S if and only if there exists a 2-form T such that (SD) o J= SW. Lemma 4.7 Let (M, g) be a compact orientable Riemannian manifold and for a vector field V, let v(X) = g(V,X). Then fMVzOidVg = 0 for every closed 1 -form 8 if and only if v = SW for some 2-form T. fm viXidVg = 0 for every vector field X if and only if v = 0.

Proof. Taking 8 exact, say df, we have (v, df) = 0 and hence (Sv, f) = 0 for every smooth function f where (.,.) denotes the global inner product of differential forms. Therefore Sv = 0 so that v = w + SW for some 2-form ID and harmonic 1-form w. Now taking 0 = w, (v, 8) = 0 gives (w, w) = 0. Thus w = 0 and v = SW. Conversely for v = SW and 0 closed, (v, 8) = (SW, 8) = (W, d0) = 0. For the second statement we already have v = SI but now let X be the contravariant form of ST, then 0 = fm viXidVg = (SW, SW) giving

v=0. We now look again at the diffeomorphism group, Di f f, of M. For a vector field X on M, let ft be its 1-parameter subgroup in Di f f . Then ft (m) is the integral curve of X starting at m E M, so in particular

dtft(m)I t=o =

X(m).

Conversely given a path ft in Di f f with fo = id, we have for every m E M, dt ft(m) I t=o E T,,,,M. Thus the tangent space to Di f f at the identity may be viewed as the Lie algebra of vectors fields, X, on M. Now consider the orbit of 0 of g E M under Di f f We have remarked (Section 4.3) that the tangent space T9M is the set of symmetric tensor fields of type (0, 2) on M and ask what are the symmetric tensor fields that are tangent to the orbit. Let z,ig : Di f f -i M be defined by 'cb9 (f) = f *g. Then Og* : I -- TOg, so given X E I, let ft be its 1-parameter subgroup. Then .

09*(X) =

dtf*

g = Cxg.

Suppose now that a symmetric tensor field D E TgM is orthogonal to (9g at g. Then

0 = (-'xg,D) =

fM

(ViXj +VjXi)DijdVg

M Dij)XjdVg = 2 f M(ViXj)D'jdV, = -21(V

for every vector field X and hence by Lemma 4.7, SD = 0.

48

4. Associated Metrics

Now let M be a compact symplectic manifold and g an associated metric with corresponding almost complex structure J. A symmetric tensor field D is orthogonal to the orbit of g under action of S if and only if there exists a 2-form ' such that (JD) o J = 5W. To see this first note that the tangent space to the orbit of g under the action of S at g is the set of tensor fields of the form £xg where X is a symplectic vector field. As we have seen (Theorem 1.5) Xi = Jikek for some closed 1-form 0. Then as in the argument above (£Xg, D) = 2 f (8D)iT`k8kdVg M

and the result follows from Lemma 4.7.

4.5 4.5.1

Examples of almost contact metric manifolds R2n+1

In Example 3.2.1 we considered R2n+1 with its usual contact structure dz 1 yidxi and saw that the contact subbundle D is spanned by -mL +y2 a y;, i = 1 ... n. For normalization convenience, we take as the standard contact structure on R2n+1 the 1-form 77 = 2 (dz - E2 1 yidxi). The characteristic vector field is then = 2-2- and the Riemannian metric ii

((dxi)2+(dyi)2) i=1

gives a contact metric structure on R2n+1. For reference purposes, we give the matrix of components of g, namely 1

uij + YiYj

0

-yi

4

0

Sid

0

-yj

0

1

The tensor field 0 is given by the matrix 0

aij

0

Vii

0

0

yj

0 0

and the vector fields Xi = 2 ayt X, ,+i = 2 a + yti aZ) , i = 1,... , n and form a O-basis for the contact metric structure. The Riemannian metric given here has the following properties. The vector field is a Killing vector field, i.e., it generates a 1-parameter group of isometries. The sectional curvature of any plane section containing is equal to 1. =

4.5 Examples of almost contact metric manifolds

49

The sectional curvature of a plane section spanned by a vector X orthogonal to and OX is equal to -3; for this reason this example is often denoted R2n+1(-3).

In dimension 3 this example is often identified with the Heisenberg group 0

1

z x

0

0

1

1

HR =

y

x,y,zER

;

left translation preserves p and g is a left invariant metric on HR. We have already seen that associated metrics are not unique and in Section 7.2 we shall give another associated metric of the contact form 77 on R 2n+1 which is less standard but which has some interesting and basic properties. .4.5.2

M2n+1 C M2'n+2 almost complex

We begin with a result of Tashiro [1963] that every C°° orientable hypersurface of an almost complex manifold has an almost contact structure. Let (M2n+2, J)

be an almost complex manifold and t : M2n+1 --, M2n+2 a C°° orientable hypersurface. There exists a transverse vector field v along M2n+1 such that Jv is tangent. For if Jt*X is tangent for every tangent vector X, Jt*X = t* f X defines a (1,1)-tensor field f on M2n+1. Applying J we have f2 = -Ion M2n+1 making M2n+1 an almost complex manifold, a contradiction. Thus there exists a vector field on M2n+1 such that v = Jt* is transverse. Define a tensor field 0 of type (1,1) and a 1-form rl on M2,+' by

Jt*X = t*qX +,q(X)v;

(*)

then applying J we have

-t*X = c*q2X + r/(cX)v - rl(X)L* and hence 02 = -I + 77 ® and rl o 0 = 0. Taking X =

v = L*o +

in equation (*) gives

and hence c = 0 and rl() = 1. Therefore (0, , rl) is an

almost contact structure on M2n+1 If M2n+2 is almost Hermitian with Hermitian metric g, set g = t*g and take v to be a unit normal. Then JP is tangent and defines by Jv Then again using equation (*)

9(X, Y) = 9'(Jt X, Jt*Y) = g(OX, OY) + rl(X)rl(Y) and we see that (0, , 77, g) is an almost contact metric structure. We can construct the usual contact structure on an odd-dimensional sphere in this way. Let Stn+1 be a sphere of radius r in C2n+2 with its usual Kahler

4. Associated Metrics

50

structure denoted as on Man+2 as above with V denoting the connection on C2n+2. With v as the unit outer normal, rl is the standard contact form (cf. Example 3.2.3). Since S?n+l is an umbilical hypersurface, its second fundamental form is o-(X, Y) = - rg(X, Y)v. Thus, using the fact that J is parallel and the Gauss-Weingarten equations, we have 0

= (oxJ) = oxv - J(Vxs + o-(X,

)) =

rX - oox - ,-.77(X)

where V is the Levi-Civita connection of g. Applying This in turn yields dr7(X, Y)

Thus for r

we have Vx -rOX.

= 2 (g(Vxe, Y) - g(VYS, X)) = rg(X, OY).

1 g is not an associated metric, but this situation is easily a contact metric =

r

rectified. The structure ? = 171,

structure. Alternatively the metric g' (1 - )77 ®r7 is an associated metric for the induced contact form y on STn+i Of course in general one cannot expect the induced almost contact metric structure to be a contact metric structure. The condition for this when the ambient space is Kahler was obtained by Okumura [1966] and we have the following theorem.

Theorem 4.12 Let Men+' be a hypersurface of a Kahler manifold Men+2, (0, e, 77, g) its induced almost contact metric structure and A its Weingarten map. Then (0, , 77, g) is a contact metric structure if and only if A0 + 0A =

-20. Proof. From the Gauss-Weingarten equations we have on the one hand 17x = Vx + o(X, ) and on the other

Vx _ qAX. Therefore 2dr7(X, Y) = g(V

Y) - g(V y X) = g((OA + Ac)X, Y) ,

from which the result follows. 4.5.3

S5 C S6

First consider R' as the imaginary part of the Cayley numbers ® and define a vector product u x v by the imaginary part of uv. Then

(uxv)

x (vxw)+(u-w)v-(u.v)w

4.5 Examples of almost contact metric manifolds

both sides not being

51

0 as in dimension 3. Also

though again in dimension 7, this would not hold if the second u were replaced by a fourth vector. The unit sphere (S6(1), g) in R7 with outer unit normal N inherits an almost complex structure J defined by JX = N x X. From the above vector identities

J2=Nx(NxX)=-X, g(JX, JY) = (N X X) (n x Y) = X Y = g(X,Y) giving S6 an almost Hermitian structure (J, g). One can also show that

(OxJ)Y + (DYJ)X = 0, and hence that the almost Hermitian structure is nearly Kahler (see also example 6.7.3 below).

Now consider the totally geodesic 5-sphere in S6(1) C R' defined by x' = 0 . 77, g) be the induced almost contact metric structure; in particular 6 a is a e= -Jv=Nx ax' x x axi ax'

with v = - a9 Let

=X1 a ax6

a -X3 a - x 2ax5

x4

a a + x5axe a - x 6ax1.

ax4 + axi 7 is the restriction of x'dx6 - x6dx1 + x5dx2 - x2dx5 + x4dx3 - x3dx4 to S5 and hence is the usual contact form. Compare this with the construction of (0',

, rl, g)

on

S5 C R' (X7 = 0) ^' C3 = {x2 + 2x5, x3 + 2x4, x6 + ix1};

J'a82 = aa5, etc., so viewing c3 C C4 c 0, J' is just left multiplication by aa, considered as an imaginary unit in G. Then for X J_

X YX = ax7X 19

ax'

xX

since aa, I X. Then g(q X, O'X) = (N x X) (aa, x X) = 0. Therefore 77, g) is an almost contact metric structure with i contact and its characteristic vector field but is not a contact metric structure, dr7(X,Y) = g(X, ci'Y) 54 g(X, cY). The difference between 0 and 0' will be seen again in Example 6.7.3 by comparison of their covariant derivatives.

4. Associated Metrics

52

The Boothby-Wang fibration

4.5.

-b

M2n the Let M2n+1 be a compact regular contact manifold and it : M2n+1 Boothby-Wang fibration of M2n+1 over a symplectic manifold M2n of integral

class with symplectic form ft Let G be an associated metric to St and J the corresponding almost complex structure; in particular (M2n, J, G) is almost Kahlerian. As we have seen the contact form 77 can be viewed as a connection form on the principal circle bundle M2n+1 Thus denoting the horizontal lift

by -r, we define a tensor field ¢ on M2n+1 by OX = RJ7r*X. Then, since the characteristic vector field is vertical, 02 = -I + 77 ® and (0, , r7) is an almost contact structure. Now define a Riemannian metric on M2n+1 by g = 7r*G +,q ®ii. Since di7 _ 7r*Sl we have

g(X, OY) = G(7r*X, J7r*Y) o it = 52(7r*X, 7rY) o It = it*Sl(X, Y) = drl(X,Y).

Clearly r7(X) = g(X, ) and hence (0, g) is a contact metric structure on M2n+1 It is also clear that e is a Killing vector field, i.e., generates a 1-parameter group of isometries. A contact metric structure for which the characteristic vector field is a Killing vector field, is called a K-contact structure, a notion that we will discuss further in later chapters.

We can at this point give a topological result on compact regular contact manifolds. Since the characteristic class of the principal circle bundle it : M2n+1) M2n is [S1] E H2(M2n, Z), the bundle is non-trivial and hence the Gysin sequence becomes 0

-

Hl(M2n R)

H1(M2n+1 R) , H°(M2n R) - H1(M2n R) , .. .

where L is left exterior multiplication by Q. Thus L is an isomorphism of H°(M2n, R) into H2(M2n,R) and therefore the map ir* is an onto isomorphism giving the following theorem of Tanno [1967a].

Theorem 4.13 Let it : M2n+1 ---) M2n be the Boothby-Wang fibration of a compact regular contact manifold M2n+1 Then the first Betti numbers of M2n+1 and M2n are equal.

In Example 3.2.7 we saw that the 5-dimensional torus carries a contact structure; we note however that it is not regular. Theorem 4.14 No torus T2n+1 can carry a regular contact structure. Proof. If T2n+1 admitted a regular contact structure, it would be a principal circle bundle over a symplectic manifold M2n by the Boothby-Wang fibration. We have just seen that the first Betti number of the base b1(M2n) is equal to b1(T2n+1) = 2n + 1. On the other hand we have the homotopy sequence of the bundle 0 - it2(M2n) > ir1(Sl) -* 71 (T2n+1) 7rl(M2n) -> 0

4.5 Examples of almost contact metric manifolds

53

since 72 (T 2,,+1) = 0. Now consider the universal covering space R2n+1 of Ten+1

and the lift of the fibration; each circle lifts to a line and hence the fibration of T2'n+l by circles has no null-homotopic fibres. Thus the map from ¶1(S') into 71(T2n+1) is non-trivial and hence 9r2(M2n) = 0. Then, by the exactness of the sequence, 7r1(M2n) = z®®z and hence b1(M2'n) = 2n, a contradiction. 11

A proof that no torus can carry a K-contact structure can be found in Itch [1997].

4.5.5 Men x R

Let (M2n, J, G) be an almost Hermitian manifolds with local coordinates x1, ... , x2n and let t be the coordinate on R. Then on Mgr` x R set rl = f dt, =fat for some non-vanishing function f. Note that dr1 = df A dt and so ri A drl 0. Without stressing notation for simplicity, set g = G + rl ® rl and define

by O = 0 and OX = JX for X orthogonal to . Then (0, , ri, g)

is an almost contact metric structure which is certainly not a contact metric structure. Generally in this example f is taken to be identically 1. However since for a contact metric structure the integral curves of are geodesics, as we have seen, the question of whether for an almost contact metric structure the integral curves of must be geodesics sometimes arises. That the integral curves of need not be geodesics can be seen in this example by choosing f so that it is not independent of xi. For then, using the standard formula for the Levi-Civita connection, 2g(VxY, Z) = Xg(Y, Z) + Yg(X, Z) - Zg(X, Y) + g([X,Y], Z) +g([Z,X],Y) - g([Y, Z], X), we have 2g(V , a

i)=g([a

)-g([E,

.if) t,e)=- faxi.

4.5.6 Parallelizable manifolds

Let Men+1 be a parallelizable manifold with {X1, ... , X2,,+1} a set of parallelizing vector fields. Define a Riemannian metric by g(XA, XB) = JAB. Let = X2n+1 and rl its covariant form with respect to g. Similarly let wi be the covariant form of Xi, i = 1, ... , n and wi' that of Xi. = Xn+i. Then define by n

=,-(w2

®Xi. - w2µ 0 Xi)

i=1

and it is easy to check that (0, , ri, g) is an almost contact metric structure. In particular any odd-dimensional Lie group carries an almost contact structure.

5

Integral Submanifolds and Contact Transformations

5.1

Integral submanifolds

Let Men+1 be a contact manifold with contact form 77. We have seen that = 0 defines a 2n-dimensional subbundle D called the contact distribution or subbundle and that since y A (d77)n 0, D is non-integrable. This nonintegrability was easily visualized, for example, in Example 3.2.6. A submanifold Mr of Men+1 is called an integral submanifold if 77(X) = 0 for every tangent vector X. It is clear that for any pair of tangent vector fields we have

d,7(X,Y) =

2(X77 (Y) -Yq (X) -71([X,Y]) = 0.

Then in terms of associated metrics, g(X, OY) = 0 and for this reason integral submanifolds are often called C-totally real submanifolds In particular 0 maps tangent vectors to normal vectors; also since l; is a normal vector, the .

dimension r can be at most n. On the other hand, by the Darboux theorem

... , yn, z) with respect to which 77 = dz - E yidxi. So xi = const., z = const. define an n-dimensional integral

we have local coordinates (x1, ... , xn, y', 1

submanifold and we have the following theorem.

Theorem 5.1 Let Men+1 be a contact manifold with contact form 77. Then there exist integral submanifolds of the contact subbundle D of dimension n but of no higher dimension.

Continuing this theme we have the following result of Sasaki [19641.

Theorem 5.2 Let (xi, yi, z), i = 1, ... , n be local coordinates about a point m = (xo) i, yo, zo) such that 77 = dz -

En 1 yidxi on the coordinate neighbor-

56

5. Integral Submanifolds and Contact Transformations

hood. In order that r linearly independent vectors Xa, .A = 1,... , r < n at m with components (aa, ba, ca) be tangent to an r-dimensional integral submanifold, it is necessary and sufficient that rl(XA) = 0 and drl(XA, Y,.) = 0, that is i i i i i i ca = E; yoaa, and Zi aabk = Ei aµba Proof. Again the necessity is clear. To prove the sufficiency, set cx,, = Ei aab' and choose a sufficiently small neighborhood U of the origin of R' with coordinates (u1, ... , Ur) such that

defines a mapping c of U into M"+'. Then axa = aa, aya = ba and au au az

= cx +

cx,,uµ

aua

i axi aua

yo

+

axi ayi u aua au" =

y

i axi aua

A

and hence the mapping t defines an integral submanifold of D tangent to Xl,... , Xr and m. Finally as in the symplectic case we note the abundance of integral submanifolds of D; more precisely we have the following result (Sasaki [1964]).

Theorem 5.3 Given a vector X E D at m E Men+1 and any r, 1 < r < n, there exists an r-dimensional integral submanifold M' of D through m with X tangent to Mr.

The proof is again immediate from the Darboux theorem, choosing the Darboux coordinates (xi yi, z), i = 1, ... , n such that X = ayl (m). In Chapter 1 we discussed a theorem of Weinstein (Theorem 1.4) that locally a symplectic manifold is the cotangent bundle of a Lagrangian submanifold. We now state an analogous theorem due to Lychagin [1977] (see also KrieglMichor [1997, p. 468]). Recall the Liouville form 0 on a cotangent bundle (Section 1.1, Examples 3.2.4, 3.2.5).

Theorem 5.4 If L is an n-dimensional integral submanifold of a contact 77), then there exists an open neighborhood U of L in Men+1, manifold an open neighborhood V of the zero section in T*L x R and a diffeomorphism (M2,+1

f :U --> V such that f IL is the identity on L and f * (0 - dt) = rl.

5.2 Contact transformations

5.2

57

Contact transformations

Recall that a diffeomorphism f of Men+1 is a contact transformation if f *77 =

T77 for some non-vanishing function r and that f is a strict contact transformation if r - 1. Clearly f *drl = dr A r) + Td77 so if d77 is invariant, (r - 1)d77 = -dr A 77 and hence (r - 1)77 A d77 = 0 giving r - 1. Thus f is strict if and only if d77 is invariant.

Theorem 5.5 f is a contact transformation if and only if X E D implies f*X E D.

Proof. r7(f*X) _ (f*7l)(X) = r77(X) = 0 and conversely 0 = 77(f*X) _ (f *77) (X) implies that f *,q is proportional to 77.

Theorem 5.6 A diffeomorphism f is a contact transformation if and only if f maps r-dimensional integral submanifolds to r-dimensional integral submanifolds.

Proof. If f is a contact transformation and MT an integral submanifold, then for X tangent to M', f*X E D by the previous result and therefore f (MT) is an integral submanifold. Conversely given X E Dm, we have seen there exists an integral submanifold MT through m with X tangent to Mr. Now since f (MT) is an integral submanifold, f*X E D and hence f is a contact transformation.

If £X77 = arl for some function u, X is called an infinitesimal contact transformation. If v - 0 we say that X is a strict infinitesimal contact transformation.

Theorem 5.7 A vector field X is an infinitesimal contact transformation if and only if there exists a function f on Men+1 such that X = -1OV f + f Proof. For the sufficiency we compute as follows. (-Ix7)(Y) = (X-j d77) (Y) + df (Y) = 2d77(-1 OVf + fe, Y) + Yf

_ -d77(oV f, Y) + Yf = -9(V f, Y) + 77(Df)77(Y) + Yf = (f)77(Y) Conversely £xr7 = Q77 implies X77(Y) - 77([X, Y]) = a77(Y). Then setting f =77(X) we have 2d77(X,Y) + Yf = Q77(Y) or

Y) -2g(OX, Y) + g(Vf, Y) = and applying 0 we have from which -20X + V f =

58

5. Integral Submanifolds and Contact Transformations

X1OVf+ as desired; note also that o _ f and hence we have the following corollary.

Corollary 5.1 X is strict if and only if f = 0. As in the symplectic case (Theorem 1.6) we have the following theorem of Hatakeyama [1966].

Theorem 5.8 Let Men+1 be a compact contact manifold. Then for any two points P, Q, there exists a contact transformation mapping P to Q. If M is regular, there exists a strict contact transformation mapping P to Q.

Proof. We first prove the result for a Darboux neighborhood U about P = (0, 0, 0). Suppose Q = (A', Bt, C) in this coordinate system and define a function f on U by f = (Bixi -A'y') +C-1 E AtBt. Let X be the infinitesimal contact transformation generated by f (strictly speaking X is determined by f E C°°(M) such that on U, f is as given and f vanishes outside some larger neighborhood). Writing x = X i8 + Xi* ya ; + X '' 9 , we have since f = 0 on U, di (X, dri(X,

a 1 a dr7(2OVf + fS, axi) _ -2g(OVf, 0axi) axi) = a

-?7(Vf)77(ax%))

2(g(Vf, axi) Xi.

-2 ax + 2(M77( axi) = -2B'

= Bt. Similarly drl(X, ey;) = 2At giving Xi = At. Now f = EA''Bi. r7(X) = X°-yiXi = X°-yiAi and so X°-yiAi = Bixi-yiAi+C-2 and hence Therefore

X _ A' axi

+ Bi a + (Bixi + C - 2 y ii

A'Bi) az

Thus the integral curves of X in U are given by xi = Ait, yi = Bit, z = (C - a AiBi) t + (E A'B') 2 When t = 1 this curve is at Q. Thus the .

corresponding 1-parameter group ft of X gives a diffeomorphism fl mapping P to Q. Now for P and Q in Men+1, the usual continuation argument gives the result. Now for M regular, the result is a consequence of the following lemma.

Lemma 5.1 If Men+1 is a compact regular contact manifold and f a C°° function on a neighborhood U of P such that f = 0, then there exists f E C°° (M) such that t; f = 0 and on some neighborhood V of P, f =- f .

5.3 Examples of integral submanifolds

59

Proof. Let it : Men+1 _, Men be the principal circle bundle structure of M2r`+1. Since f = 0, f is constant on fibres and therefore there exists a neighborhood V about ?r(P) and a function f on V such that f = f' o 7r on ir-1(V'). Now extend f' to a C°° function f' on Men agreeing with f' on V. Then setting f = f' 0 it we have the desired function. As in the symplectic case, Boothby [1969) mapped k points to k points; see also Kriegl-Michor [1997, p. 472. 5.3

Examples of integral submanifolds

One can readily cite some simple examples of integral submanifolds, e.g., an n-dimensional integral submanifold in R2n+1 given by yi = 0, as already noted, and the fibres of Tl M and T1M with the contact structures given in Example 3.2.4. We give a few more examples here and we will give further discussion of integral submanifolds and additional examples from time to time. 5.3.1

Sn C Stn+1

Consider the space Cn+1 of it + 1 complex variables and let J be its usual almost complex structure. Let Stn+i = {z E C'+n+1 I IzI = 11. Then as we have seen we can. give S2n+1 its usual contact structure as follows. For every _ -Jz and OX is the tangential part of z E S2n+1 and X E TZS2n+1

JX. Let g be the standard metric on S2n+1 and i the dual 1-form of . Then (0, , i , g) is a contact metric structure on Stn+i Now let L be an (n + 1)dimensional linear subspace of Cn+l passing through the origin and such that JL is orthogonal to L. Then since is simply the action of J on the position vector, Sn = S2n+1 n L is orthogonal to and is therefore an n-dimensional integral submanifold of the contact structure on S2n+1. Clearly Sn is a totally geodesic integral submanifold. 5.3.2

T2 C S5

The following imbedding of a 3-torus into the unit 5-sphere as a flat minimal submanifold is well known. Given in terms of its position vector x : T3 -3 SS C E6 : C3, it is x = 3 (cos u, sin u, cos v, sin v, cos w, sin w).

Now imbedding T2 in T3 diagonally by u + v + w = 0 we have a flat minimal surface in S5 given by

x = (cos u, sin u, cos v, sin v, cos(u + v), - sin (u + v)). 3

60

5. Integral Submanifolds and Contact Transformations

As we have seen the characteristic vector field on S5 is given by

e_ -Jx = 3 (- sin u, cos u, - sin v, cos v, sin(u + v), cos(u + v)).

Computing the tangent vectors xu = a and x, = a directly, it follows easily that < , xu >=< , x, >= 0 and hence that this torus in an integral surface of the contact structure on S5. Examples 5.3.1 and 5.3.2 show that S5 contain both S2 and T2 as integral submanifolds. It is known for topological reasons (see e.g., Steenrod [1951, p. 144]) that S5 does not admit a continuous field of 2-planes. Thus S5 cannot be foliated by integral surfaces of its contact structure. 5.3.3 Legendre curves and Whitney spheres

Recall that a 1-dimensional integral submanifold of a contact manifold is called a Legendre curve and we begin with an elementary property of Legendre curves

in the contact manifold (R', ,q = dz - ydx). The projection y* of a closed Legendre curve y in R3 to the xy-plane must have self-intersections; moreover the algebraic (signed) area enclosed by y* is zero. Since dz - ydx = 0 along -y, this follows from the elementary formula for the area enclosed by a curve given by Green's theorem,

0=-

J

7

dz = j -ydx = area, 7

the area being + for -y* traversed counterclockwise and - for clockwise. Legendre curves and their projections are discussed further in Section 8.3. Here we note that one can think of the pair of 'y and its projection y* in the following terms. Suppose that y itself does not have self-intersections and regard y* as a Lagrangian submanifold in R2 = C with self-intersections; then think of going from y* to y as a way of removing the singularity but preserving the "Lagrangian-Legendre" property. For example the map of the circle u2 + v2 = 1 into R2 given by (u,v) -) (v,2uv)

has a double point, viz., (±1, 0) -> (0, 0). On the other hand the map of the circle u2 + v2 = 1 into (R3, y = dz - ydx) given by (u, v) --> (2uv, v, 2u - 3u3) is an imbedding and is a Legendre curve.

5.3 Examples of integral submanifolds

61

A generalization of this example and an important Lagrangian submanifold of R2n = Cn is the Whitney sphere. We give two descriptions of the Whitney sphere. Let S2 = F-1, dxi A dyi be the standard symplectic form on R2n and consider the sphere Sn in Rn+1 given by El °(ui)2 = 1 immersed in R2n by (u°,

... un) -.' (ul, ... un, 2u°ul,... ,

,

,

2u°un).

Again notice the double point (±1, 0,... , 0) and it is easy to check that this immersed sphere is a Lagrangian submanifold of R2n (cf. Weinstein [1977, p. 26], Morvan [1983]). In Section 1.2 we remarked that the sphere Sn can not be imbedded in Cn as a Lagrangian submanifold. In a related vein there are no umbilical, non-totally-geodesic, Lagrangian submanifolds isometrically immersed in any complex space-form (Chen-Ogiue [1974b]). Now imbed E °(uz)2 = 1 in the contact manifold R 2n+' by (2u°ul

un)

(u°

un 2u° - (u°)3)

2u°un ul

3 giving an imbedded sphere as an integral submanifold of the standard contact structure. We refer to this sphere as a contact Whitney sphere .

The Whitney sphere is often presented in another form, which, though slightly more complicated, lends itself to a natural geometric characterization. For the Whitney sphere Mn as a Lagrangian submanifold of R2n - Ctm, the immersion is (u0

u)n)

1

1 n O On 1+(u°)2(u,...,u,uU11 ...,uu).

This submanifold satisfies the relation

n+2 H2- n2-(n-

T

where H is the mean curvature vector and r the scalar curvature of Mn. This equality characterizes the Whitney sphere as a Lagrangian submanifold of Cn. More precisely it was proven by Borrelli, Chen and Morvan [1995]

and independently by Ros and Urbano [1998] that if Mn is a Lagrangian n +2 submanifold of Cn, then IHI2 > J)'T with equality if and only if Mn

- n2(n -

is either totally geodesic or a (piece of a) Whitney sphere. Borrelli, Chen and Morvan [1995] and Ros and Urbano [1998] also gave the characterization that the second fundamental form o of a Lagrangian submanifold in Cn is given by U(X'Y)

n+2(g(X,Y)H+g(JX,H)JY+g(JY,H)JX)

5. Integral Submanifolds and Contact Transformations

62

if and only if the submanifold is either totally geodesic or a (piece of a) Whitney sphere. In the contact manifold R2n+1 with its standard contact metric structure (Example 4.5.1) we also have a second presentation of the contact Whitney

sphere as an imbedded sphere and an integral submanifold of the contact structure, namely (4G

0

46

n

)

1

0

Cu u 1 + (4.60)2

1

...

On

, 4L 26

4L

1

n U,

U0

1

1 + (4.60)2 /

For this contact Whitney sphere the analogues of the above results of Borrelli, Chen and Morvan, and Ros and Urbano were given by A. Carriazo and the author [toap]. 5.3.4

Lift of a Lagrangian submanifold

Let 7r : Men+1 ___+ M2n be the Boothby-Wang fibration of a compact regular contact manifold Men+1 over a symplectic manifold Men of integral class

with symplectic form 0 and recall the details of Example 4.5.4. Let L be a Lagrangian submanifold of Men and consider the set of fibres over L. Then since OX = i'rJrr,,X, this is a submanifold Nn+1 of Men+1 with the property that 0 maps the tangent space into the normal space; such a submanifold is sometimes called an anti-invariant submanifold. If X and Y are horizontal tangent vector fields to Nn+1 then 0 = 29(X,gY) = 2d97 (X, Y) = X77 (Y) -Yrl(X) - rl([X,Y]) = -'([X,Y])

Thus the horizontal distribution in Nn+1 is integrable giving n-dimensional integral submanifolds of Men+1

6 Sasakian and Cosymplectic Manifolds

6.1

Normal almost contact structures

Recall that almost contact manifolds were defined as manifolds with structural group U(n) x 1 and hence can be thought of as odd-dimensional analogues of almost complex manifolds. We now consider almost contact manifolds which are, in the sense to be defined, analogous to complex manifolds. As is well known an almost complex structure need not come from a complex structure. The celebrated theorem of Newlander and Nirenberg [1957] states that an almost complex structure J of class C2n+, with vanishing Nijenhuis torsion is integrable, i.e., is the corresponding almost complex structure of a complex structure. The Nijenhuis torsion [T, T] of a tensor field T of type (1,1) is the tensor field of type (1,2) given by [T, T] (X, Y) = T2 [X, y] + [TX, TY] - T [TX, Y] - T [X, TY].

All manifolds under consideration are of class C°°, so the theorem of Newlander and Nirenberg applies. For detailed studies of complex manifolds see for example Goldberg [1962], Kobayashi and Nomizu [1963-69, Chapter IX], Kobayashi and Wu [1983], Morrow and Kodaira [1971], Yano [1965].

Let Men+i be an almost contact manifold with structure tensors

77)

and consider the manifold Men+i x R. We denote a vector field on Men+1 x R

by (X, f A) where X is tangent to Men+i t the coordinate on R and f a C°° function on Men+i x R. Define an almost complex structure J on Men+i x R by

J(X, f dt) = (OX - f , 71(X) dt

64

6. Sasakian and Cosymplectic Manifolds

that P = -I is easy to check. If now J is integrable, we say that the almost contact structure (0, , ri) is normal (Sasaki and Hatakeyama [1961]). As the vanishing of the Nijenhuis torsion of J is a necessary and sufficient condition for integrability, we seek to express the condition of normality in terms of the Nijenhuis torsion of 0. Since [J, J] is a tensor field of type (1,2), it suffices to compute [J, J]((X, 0), (Y, 0)) and[J, J]((X, 0), (0, at)) for vector fields X and Y on Men+1

[J, J]((X,0), (1' 0)) _ -([X,Y],0)+

-J

[(x(x))

(Y, o)]

[(x (X))

-1 I(X,O),

(Oy"q(y)dt)1 ,

(y(Y))]

= (O2[X,Y] - ([X, Y]), 0) + [OX, OY], (OX91(Y) - OY?7(X ))d)

- (0[OX,Y]+(Yq OY])d)

(O[X, OY] -

_

([](xY) + 2d(X,Y), ((-1Ox?7) (Y) - (_OYrl)(X))d) [J, J]((X, 0), (0, dt))

_ [(OX, (X)), (-c, 0)]

-J [(OX, 77(X)- )'(0, d )]

- J[(X, 0), (-i, 0)]

_ (-[OX, ], (671(X ))d) + (0[X, 6]"q ([X, 0 d

)

_ ((£O)X, Vo)(X )). We are thus lead to define four tensors N('), N(2), N(3), N(4) by N(') (X, Y) = [0, 0] (X, Y) + 2dr7(X, N(2) (X,Y) _ (£Oxii)(Y) - (SOY77) (X), N(3)

N(4)

= ('CO)X,

=

(_p77)(X).

6.1 Normal almost contact structures

65

Clearly the almost contact structure (0, , rl) is normal if and only if these four tensors vanish. However the vanishing of N(' implies the vanishing of N(2), N(3) and N(4), so that the normality condition is simply [0, 0] (X, Y) + 2drl(X, Y) e = 0.

We now prove this and other properties of these tensors (cf. Sasaki and Hatakeyama [1961], [1962]).

Theorem 6.1 For an almost contact structure

rl) the vanishing of Nlli

implies the vanishing of N(2), N(3) and N(4).

Proof. Setting Y =

and recalling that drl(X, ) = 0, we have

0 = [0,0](X,0 = 02[X,e] -O[OX,e] = 0((_'EO)X).

Applying 0 and noting that drl(e, OX) = 0 implies rl([l;, OX]) = 0, we have 0, but N131 = 0. Moreover 0 is immediate and hence N(4) = 0. Finally applying rl to 0 = [0, 0] (OX, Y) + 2dr7(gX, Y)1=

we have rl([02X, OY])+OXrl(Y) -rl([OX, y]) = 0 which simplifies to

AT(2)

= 0. 171

Theorem 6.2 For a contact metric structure (0,1 rl g), N(2) and N(4) vanish. Moreover N13> vanishes if and only if

is a Killing vector field.

Proof. We have already seen that N(4) = 0. Now N(2) can be written N(2) (X, Y) = 2dr7(OX, Y) - 2dr7(OY, X) = 2g(OX, OY) - 2g(gY, OX) = 0.

Turning to NO), we note that since drl is invariant under the action of

0 = (' drl)(X,Y) = g(X,OY) -

X], OY) - g(X,

(,o)(X, OY) + g(X, (Zq)Y) from which we see that NO3) = 0 if and only if 1; is Killing.

Next we establish a formula for the covariant derivative of 0 for a general almost contact metric structure (0,1=, rl, g).

Lemma 6.1 For an almost contact metric structure

rl, g), the covariant

derivative of ¢ is given by 2g((Vxql)Y, Z) = 3d(D(X, OY, OZ) - 3dc(X, Y, Z) + g(N(l)(Y, Z), OX) +N(2) (Y, Z)rl(X) + 2dq(gY, X)77(Z) - 2drl(gZ, X)rl(Y).

66

6. Sasakian and Cosymplectic Manifolds

Proof. Recall that the Levi-Civita connection 17 of g is given by

2g(VxY, Z) = Xg(Y, Z) + Yg(X, Z) - Zg(X, Y)

+g([X,Y],Z)+g([Z,X],Y) -g([Y,ZI,X) and that the coboundary formula for d on a 2-form d) is d,D(X,Y,Z) = 3{X.T.(Y,Z)+Y.D(Z,X)+ZD(X,Y)

-`D([X,YI, Z) - D([Z,X],Y) - 1([Y, Z],X)}. Therefore

2g((Vxo)Y, Z) = 2g(VXOY, Z) + 2g(7xY, OZ)

= Xg(OY, Z) + oYg(X, Z) - Zg(X, OY)

+g([X,OY],Z) +g([Z,X],OY) - g([OY, ZI,X) +Xg(Y, qZ) + Yg(X, OZ) - cZg(X, Y) +9([X, Y], cZ) + g([OZ, X], Y) - g([Y, OZ], X)

= X b(Y, Z) +lY(Ccbz, X) +77(Z)77(X)) - ZD(X,Y) -(D([X, OY], OZ) + r7([X, oY])r7(Z) +ID([Z, X], Y) - gWgY, Z], OX) + i7(X)77([Z, OY] )

OZ) - Y(D(Z, X) - OZ(CgY, X) + 71(Y)71(X)) Y], Z) - "'([OZ, XI, OY) +77([cZ, X])77(Y) -g(o[Y, OZ], OX) + r7(X)r7([OZ, Y])

+ {ID([Y, Z], X) - 9([Y, Z], OX) }

- {D([qY, 0Z], X) - g([OY, OZ], OX) } + {g(2dr1(Y, Z) , OX)}

= 3d(b (X, ¢Y, OZ) - 3db(X, Y, Z) + g(N(' (Y, Z), OX)

+N(2)(Y,Z)7i(X) +2d7(cY,X)77(Z) - 2d?7(gZ,X)71(Y).

Corollary 6.1 For a contact metric structure the formula of Lemma 6.1 becomes

2g((VxO)Y, Z) = g(N(1) (Y, Z), OX) + 2dr1(OY,

2d?7(OZ,

X)(Y).

6.2 The tensor field h

67

Taking X = e in Corollary 6.1 we see that 1760 = 0 for any contact metric structure. By choosing a 0-basis, Corollary 6.1 also yields V2O'j = -2nr7j on a contact metric manifold. While our main interest is in contact manifolds, we mention, in regard to the normality of almost contact structures, papers of Sato [1977] and Geiges [1997b]. Sato proved that if a compact 3-dimensional normal almost contact manifold M is not homotopic to Sl X S2, then ire (M) = 0 and Geiges gives a complete classification.

6.2

The tensor field h

We have seen that on a contact metric manifold, N(3) vanishes if and only if is Killing (Theorem 6.2) and a contact metric structure for which is Killing is called a K-contact structure . For a general contact metric structure the tensor field N(3) enjoys many important properties and for simplicity we define a tensor field h on a contact metric manifold by h = 2160 = 2N(3)

The first property to note is immediate, namely ht; = 0. We now give a number of important properties of h. Lemma 6.2 On a contact metric manifold, h is a symmetric operator,

vx, = -OX - OhX, h anti-commutes with 0 and trh = 0.

Proof. We have already seen that on a contact metric manifold, V

=0

and V = 0. Thus 9((I q)X, Y) =

oox - oo£X +

Y)

= g(-V xe + ovxe, Y) which vanishes if either X or Y is l;. For X and Y orthogonal to ', N(2) = 0 becomes rl([OX, Y]) + i7([X, OY]) = 0; continuing the computation we then have

9(( C0)X,Y) = rl(DOxY) +77(OxgY) 77(VYOX) +77(V X)

68

6. Sasakian and Cosymplectic Manifolds

= 9((£44)Y,X) For the second statement using Lemma 6.1 we have

_

Z) = g(02[x, Z] -

OZ], OX) - 2dr7(OZ, X)

=

29(OZ,OX) 2g(Z, X) + 277(Z)r7(X)

X) + rl

_ -9((£Cq5)X, Z) - 29(X, Z) + 29(77(X), Z) Applying 0 we then have X+ and hence -OVx = -2

17x = -OX - OhX. To see the anti-commutativity note that 29(X, OY) = 2dr7(X, Y) = 9(V

Y) - 9(V y , X)

= g(-OX - OhX,Y) - g(-OY - ghY,X). Therefore 0 = g(-ghX,Y) - g(Y, hcX) giving hq + Oh = 0. An immediate consequence of this anti-commutativity is that if hX = AX, then hqX = -AqX, thus if A is an eigenvalue of h so is -A and hence trh = 0. As a result we get the following easy corollary.

Corollary 6.2 On a contact metric manifold, br7 = 0. Proof. Choosing an eigenvector basis {e;,} of h we have ez) = g(-Oei - Ohe2, ei) = 0.

Example 3.2.6 provides a nice illustration of the tensor field h. Let g = 2(cosx3dxi + sinx3dx2) be the contact form on R3; then = 2(cosx3al + sin x3 8x2) is the characteristic vector field. The flat metric g2j = S2, is an associated metric and 0 is given by

l=

a

0 0

0

0

sin x3 - Cos x3

- sin x3

Cos x3

0

D is spanned by X = -sin x3 8y1 + cos x3 aa2 and ae3 Since the metric is Euclidean on 1183, Vx = 0. Therefore by Lemma 6.2, h0X = OX, but OX = .

6.3 Definition of a Sasakian manifold

69

and so has = a3i i.e., aa3 spans the +1 eigenspace of h and in turn X spans the -1 eigenspace.

X3

Conditions on V h arise frequently as we will see in later chapters, but we mention a couple at this point. Calvaruso and Perrone [2000] prove that a 3dimensional contact metric manifold is locally homogeneous if and only if it is ball-homogeneous and V h = 2ahq where a is a constant. Moreover recalling the contact circles (Example 3.2.9) of Geiges and Gonzalo [1995], Calvaruso and Perrone prove that a compact orientable 3-manifold admits a taut contact circle if and only if it admits a locally homogeneous contact metric structure satisfying VEh = 0.

6.3

Definition of a Sasakian manifold

In this short section we give an important definition, namely that of a Sasakian manifold. A Sasakian manifold is a normal contact metric manifold. In some respects Sasakian manifolds may be viewed as odd-dimensional analogues of

Kahler manifolds. This point of view is reflected in many of the examples and results on Sasakian manifolds that will be discussed. To begin with, the following theorem is analogous to an almost Hermitian manifold being Kahler

if and only if the almost complex structure is parallel with respect to the Levi-Civita connection.

Theorem 6.3 An almost contact metric structure (0, l;, 77, g) is Sasakian if and only if

(VxO)Y = g(X,Y) - r7(Y)X.

70

6. Sasakian and Cosymplectic Manifolds

is a Proof. The necessity follows easily from Lemma 6.1, for if normal contact metric structure, (D = dr7, N(' = 0 and N(2) = 0, and hence 2g((Vxq5)Y, Z) = 2dr7(gY, X)g(Z) - 2dr7(gZ, X)ii(Y) = 2(g(Y, X) - r7(Y)r7(X))r7(Z) - 2g(Z, X) - rJ(Z)rl (X ))rl (Y) = 2g(g(X, Y) - 77(Y)X, Z).

Conversely, assuming (Vxo)Y = g(X, Y) - r7(Y)X, setting Y =

gives

-cVx = 77(X) - X and hence Vx = -OX. Therefore d77(X,Y) = 2

Y) -

X)) = g(X, OY)

showing that (0, , r7, g) is a contact metric structure. Now

[0, 0](X, Y) _ (OVY0 - V yo)X - (OVxO - VOXO)Y and straightforward substitution of the hypothesis simplifies this to [0, 0] (X, Y) = -2dr7(X, Y) .

Recall (Example 4.5.4) that a contact metric structure (0, , 71, g) is said to

be K-contact if 6 is a Killing vector field. Since Vxe = -OX in the above proof and 0 is skew-symmetric, we have the following corollary.

Corollary 6.3 A Sasakian manifold is K-contact. In dimension 3 the converse is true, Corollary 6.5. For K-contact structures that are not Sasakian, see Example 6.7.2. While we will see other examples of Sasakian manifolds in due course, let us

show here that the standard contact metric structure on an odd-dimensional unit sphere is Sasakian. Recall that the standard contact metric on an odddimensional sphere was exhibited in Example 4.5.2 and in particular, if the radius is 1, the constant curvature metric is an associated metric. Using the notation there and the Kahler property of C2n+2, we have

0 = (VxJ)Y = Vx(OY + 77(Y)v) - J(VxY - g(X, Y)v) = oxOY - g(X, qY)v + (X77(Y))v + 77(Y)X

-cVxY - rl (OxY)v - g(X, Y) _ (V )Y g(X, c Y))v. Taking the tangential part we see that (Vxq)Y = g(X, Y) - 77(Y)X and hence that the structure is Sasakian.

6.3 Definition of a Sasakian manifold

71

We close this section with a couple of diagrams indicating some analogies between almost Hermitian manifolds and almost contact metric manifolds. Let (M2nJ, g) be an almost Hermitian manifold and let .l denote the fundamental 2-form. Then we have the following schematic array of structures.

ds2-0

Hermitian

Kaehler

10

A

/JO

[J,J]=0 I

almost Hermitian

I

I [J,J]=0

I

dc2=0

almost Keehler

Recall that S6 carries an almost Hermitian structure that is neither Hermitian nor almost Kahler (cf. Example 4.5.3 and Example 6.7.3 below). The well known Calabi-Eckmann manifolds S2P+1 x 529+1, p, q > 1 are Hermitian manifolds (see also Section 6.6 below) which are not Kahler for the topological

reason that the second Betti number of a compact Kahler manifold is nonzero. As noted in Section 1.1 there are many compact symplectic (and hence almost Kahler) manifolds with no Kahler structure. Also the tangent bundle of a non-flat Riemannian manifold carries an almost Kahler structure which is not Kahlerian (Dombrowski [1962],Tachibana and Okumura [1962], see also Section 9.1 below). Finally there are many well known Kahler manifolds. The corresponding diagram for almost contact metric manifolds is the following, the notion of a K-contact manifold being intermediate between a contact metric manifold and a Sasakian manifold.

normal

almost contact

Sasakian

metric

N(1)=0

contact 4, =drl

metric

6. Sasakian and Cosymplectic Manifolds

72

We have already seen in Example 4.5.3 that S5 carries an almost contact metric structure which is not a contact metric structure and we will see in Example 6.7.3 that the structure is not normal. Cosymplectic manifolds as discussed in Section 6.5 are examples of normal almost contact metric manifolds which are not Sasakian. Since the first Betti number of a compact Sasakian manifold is even (including zero), the 3-dimensional and 5-dimensional tori have no Sasakian structures though they have the contact structures presented in

Examples 3.2.6 and 3.2.7. Also we will see in Section 9.2 that the tangent sphere bundles are not in general Sasakian. Finally we just noted that the odd-dimensional spheres are Sasakian and other examples are given in Section 6.7. Also in the Geiges [1997b] classification of compact 3-dimensional normal almost contact manifolds, he identifies those which are normal contact (Sasakian). A similar treatment can be found in Belgun [2000].

6.4

CR-manifolds

In this section we discuss some aspects of the relation of almost contact structures and contact metric structures to CR-structures. Let N be an ndimensional C°° manifold and TCN its complexified tangent bundle, i.e., T N = TpN ®R C _- TpN ® iTpN. Let 7-l be a C°° complex subbundle of complex dimension 1. A CR-manifold, as introduced by Greenfield [1968], of real dimension n and CR-dimension 1 is a pair (N, f) such that Rp n7{, = 0 and 7-l is involutive, i.e., for vector fields X, Y E 'H, [X, Y] E 7-l. Then there

exists a unique subbundle D of TN such that Dc = ' ® 7-1, and a unique

bundle map J : D -> D such that J2 = -I and' = {X - i,TX IX E D}. Now let (M, H) be a CR-manifold with M of real dimension 2n + 1 and ' of complex dimension n. Consider Fx the space of all covectors f E T* M such

that D C ker f . This defines a real line bundle F C T*M. If M is orientable, then F --+ M admits a global nowhere vanishing section 71 which is called a pseudo-Hermitian structure and (M,7-l, r)) is called a pseudo-Hermitian manifold. The Levi form of (M, ,q, 7-l) is defined by

L,, (X, Y) = -dr1(X, ,7Y),

X, Y E D.

(M, rl,'H) is non-degenerate if L,, is non-degenerate. In this case M has a natural volume form rl n (d7)n; thus 77 is a contact form and its characteristic is transversal to D. (M, 7-l, r)) is said to be strongly pseudo-convex if L,, is positive definite. Using the direct sum decomposition TM = D ® we may extend L,, to a Riemannian metric g,, on M, called the Webster metric, (see e.g., S. Dragomir [1995]) by 1, 0 for X E D and g,,(X,Y) = L,,(X,Y) for X, Y E D. Moreover we may extend J to a tensor vector field

6.4 CR-manifolds

73

field 0 on M by cl = 0 and OX = ,7X for X E D. Therefore a strongly pseudo convex CR manifold (M, 7-l, q), carries a contact metric structure (0, , ri, g). In [1978] Bejancu defined the notion of a CR-submanifold of an almost Hermitian manifold. A submanifold N of an almost Hermitian manifold (M, J, g) is a CR-submanifold if there exists a holomorphic (J-invariant) subbundle D with DP {0}, T,N such that Dl is totally real, i.e., JDP C TP N. Clearly, real hypersurfaces are CR-submanifolds. Let P denote the projection from TN to D and Q the projection from TN to Dl. Now since D is J-invariant, set J = JP and define a complex subbundle 7-l of TeN by 7-l = {X - i,7X IX E D}. The following lemma is immediate.

Lemma 6.3 ,7(X - iJX) = JPX - i,7JPX. Now suppose that the ambient space (M, J, g) is Hermitian, i.e., the almost complex structure J is integrable. We then have the following lemma and the theorem of Blair and Chen [1979].

Lemma 6.4 Let N be a CR-submanifold of a Hermitian manifold (M, J, g). Then for X,Y E D, Q([JX, Y] + [X, JY]) = 0.

Proof. Since M is Hermitian, [J, J] = 0 and hence

0 = [J, JJ (JX, Y) = - [JX, Y] - [X, JY] + J([X, Y] - [JX, JY]),

but [X, Y] and [JX, JY] are tangent to N, so J([X, Y] - [JX, JY]) has no D1-component. Therefore [JX, Y] + [X, JY] has no D1-component.

Theorem 6.4 Let M be a Hermitian manifold and N a CR-submanifold, then N is a CR-manifold.

Proof. Let X, Y E D. Then

[X - i,7X, Y - iJY] = [X, Y] - [JX, JY] - i [JX, Y] - i[X, JY]

= -J[JX, Y] - J[X, JY] - iP[JX, Y] - iP[X, JY] by virtue of [J, J] = 0 and Lemma 6.4. Continuing the computation using Lemma 6.4 again and then Lemma 6.3,

[X - i,7X, Y - i,7Y] _ -,7[JX, y] - ,7[X, JY] + i,72[JX, y] + i,72 [X, JY] = -,7([JX,Y] - i,7[JX, Y]) -,7([X, JY] - i,7[X, JY]) C -'H. Turning to the case of almost contact structures, consider an almost contact

manifold Men+1 with structure tensors (0, , r7). Since 02 = -1 + ri ® and O = 0, the eigenvalues of 0 are 0 and ±i each with multiplicity n; in particular

74

6. Sasakian and Cosymplectic Manifolds

0 is an almost complex structure on the subbundle D defined by 77 = 0. Thus the complexification of Dp is decomposable as D'p ®Dp where Dp = {X - icX J X E Dp} and DP = {X + i0X IX E Dp}, the eigenspaces of ±i respectively.

Lemma 6.5 q(X - icX) E DP'. Proof. O(¢(X - iOX )) = 0(i(X - iOX)) = iO(X - iqLx). We now prove a Theorem of Ianus [1972] that a normal almost contact manifold is a CR-manifold. The converse is not true and in Theorem 6.6 we will obtain a necessary and sufficient condition for a contact metric manifold to be a CR-manifold. Theorem 6.5 If (M2n+1, 0, , 77, g) is a normal almost contact manifold, then

(M2-+l, D') is a CR-manifold. Proof. First of all DIP = Dp and D'p n DP = 0; thus it remains to show that [X - iqX, Y - iqY] E D' for X, Y E D. By the normality, [0, 0] + 2dr7 ® = 0,

so that 0 = -[X, Y] + [OX, OY] - O[OX, Y] - O[X, OY].

Also since N(2) = 0, (.COxrl)(Y) - (_'0Yr7)(X) = 0 from which 77([OX,Y] + [X, OY]) = 0.

Now [X - ioX, Y - iqY] = [X, Y] - [OX, OY] - i[OX, Y] - i[X, OY] _ -O[OX, Y] - O[X, OY] + i02[0x, Y] - irl([OX, Y]) +i02 [X, OY] - irl ([X, OY] )

_ -0([OX,Y] - i4[oX,Y]) - O([X,OY] - iq[X, OY]) E D'. On a contact metric manifold M2n+1, (M2,+1 D') might be CR without the structure being normal. We present an important result of Tanno [1989] giving a necessary and sufficient condition for a contact metric manifold to be a CR-manifold. Theorem 6.6 Let (M2,,+1, 77, g) be a contact metric manifold. Then the pair (M2"+1, D') is a (strongly pseudo-convex) CR-manifold if and only if

(Oxo)Y = g(X + hX,Y) - rl(Y)(X + hX).

(*)

6.4 CR-manifolds

75

Proof. The strong pseudo-convexity refers to the positive definiteness of the Levi form

L(X,Y) = -drj(X, qjDY),

X, Y E D.

We must show the equivalence of (*) and

[X - iOX, Y - iOY] E D' for X, Y E D.

(t)

Since 10) = 0, 77([X, Y] - [OX, cY]) = 0 for X, Y E D from which we can see

that (t) is equivalent to O[X, Y] - O[OX, qY] - [OX, Y] - [X, qY] = 0 X, Y E D.

($)

Now replacing X, Y by OX, OY we have an expression for our condition in terms of general vector fields X, Y, viz.,

O[cX, qY] - q[-X + i7(X), -Y + rj(Y)]

-[-X +

OY] - [¢X, -Y + i7(Y)e] = 0.

Applying 0 and changing the sign we get [0, 0] (X ,

Y) - r([qX cY] ) ,

+0 [r7 (X

02

[rl (X

OY] + O [OX ,

Y] -

O2[X,

q(Y) ] = 0.

Expressing the Nijenhuis torsion and Lie brackets in terms of covariant derivatives and noting that -r7([OX, OY]) = 2dr1(gX, OY) = 2dr1(X,Y), we have

(0'7Yo - V yo)X - (fOV xO - V xO) tY + 2di (X, Y)

-OVOYS) -11(Y)(02Vxe -gV0xS) = 0. Now recall that since g(X, qY) = dr1(X, Y), the cyclic sum on {X, Y, Z} in g((VxO)Y, Z) must vanish. Thus taking the inner product of our condition with a vector field Z and using this cyclic sum property twice we obtain g(O(VYql)X, Z) - g(O(Vxq5)Y, Z) + g((Vxq5)Z, OY) + g((Vzq5)OY, X)

-g((DyO)Z, OX) - g((V zq)OX, Y) + 2dr1(X, Y)71(Z)

+71(X) (-

g((V

Z) + g((V

OZ)) = 0.

Also since 02 = -I + r, we have (Vxo)¢Y + 0(OxO)Y = Using this we obtain

76

6. Sasakian and Cosymplectic Manifolds

2g((Vz9l)OY,X) +rl(X)g(VOY

,

OZ) - ?J(Y)g(V x , OZ) = 0.

Lemma 6.5 then yields -(Vzq )OY = (g(qZ, Y) + g(ghZ, Y by qY, (VzO)Y = g(Z + hZ,Y)e - r7(Y)(Z + hZ) as desired. In contrast to the fact that not every 3-dimensional contact metric manifold is Sasakian, every 3-dimensional contact metric manifold is a strongly pseudoconvex CR-manifold as we see in the following corollary. Corollary 6.4 A 3-dimensional contact metric manifold is a strongly pseudoconvex CR-manifold; in particular on a 3-dimensional contact metric manifold

(Vxo)Y = g(X + hX,Y) - 77(Y)(X + hX). Proof. The left-hand side of ($) when restricted to D is just -0[0, 0] and hence it is enough to verify ($) on the basis {X, Y = OXI of D which is straightforward.

Corollary 6.5 A 3-dimensional K-contact manifold is Sasakian. Proof. Since being K-contact is equivalent to a contact metric manifold satisfying h = 0, the result follows from Corollary 6.4 and Theorem 6.3.

Remark. In the literature there is also the following definition of a CRstructure which does not include the integrability. A CR-structure on a man-

ifold M is a contact form 77 together with a complex structure J on the contact subbundle D (see e.g., Chern-Hamilton [1985]). Let (M, 77, J) be a CR-structure in this sense and set

'}l={X-i,7XIX ED}= {ZEDo1,7Z=iZ}. If the Levi form L,, is Hermitian, then (M, r7, J) is called a non-degenerate pseudo-Hermitian manifold (the condition is equivalent to the partial integrability condition: [X, Y] E De for X, Y E N). (M, 71,J) is said to be integrable if 7-l is involutive, i.e., [X, Y] E D for X, Y E D. If L7, is positive definite, (M, 77,J) is called a strongly pseudo-convex CRmanifold. So the notion, in this sense, of a strongly pseudo-convex CR-structure on M is equivalent to our notion of a contact metric structure (0, , 77, g) by the relations g = L,, +,q ®r7, J = 01 D where L,7 also denotes its natural extension

to a (0,2) tensor field on M. Using this definition of CR-structure, Theorem 6.6 can be given in the following form: Let (M, 77, g) be a contact metric manifold and (77,J) the corresponding strongly pseudo-convex CR-structure; then

(r7, J) is integrable if and only if the condition (*) of Theorem 6.6 holds. In particular (M, 77, g) is Sasakian if and only if h = 0 and (r7, J) is integrable.

6.5 Cosymplectic manifolds and remarks on the Sasakian definition

6.5

77

Cosymplectic manifolds and remarks on the Sasakian definition

There are two notions of cosymplectic structure in the literature (in fact counting "Hermitian cosymplectic" (5Q = 0) there are at least three). P. Libermann [1959] defines a cosymplectic manifold to be a (2n + 1)-dimensional manifold admitting a closed 1-form 77 and a closed 2-form (D such that 77 A (Dn is a volume

element. As before on the subbundle defined by q = 0,d) may be polarized to give an almost contact metric structure (0, , r), g) for which both 97 and the fundamental 2-form (D(X, Y) = g(X, OY) are closed. In [1967] the author defined a cosymplectic structure to be a normal almost contact metric structure (0, e, rl, g) with both r) and 1 closed. Corresponding to Theorem 6.3 for Sasakian structures we have the following result.

Theorem 6.7 An almost contact metric structure (0, , 77, g) is cosymplectic if and only if 0 is parallel. Proof. Since NM = 0 implies NM = 0, that a cosymplectic manifold satisfies OXO = 0 follows immediately from Lemma 6.1. Conversely VXq5 = 0 implies Now that dD = 0 and NM = 2dr7 N(2

=

-77([OY ])

Voy - VOY) =

OVEY) = 0;

so setting Z = l in Lemma 6.1, drl(OY, X) = 0 for all X. Therefore d77 = 0 and in turn N(1 = 0. Since drl = 0 on a cosymplectic manifold, it is clear that the subbundle defined by rl = 0 is integrable; moreover one can show that 77 is parallel. Also the projection map to the tangent spaces of the integral submanifolds, -02 = I - rl is parallel. Thus from Theorem 6.7 we see that locally a cosymplectic manifold is the product of a Kahler manifold and an interval, the complex structure being the restriction of 0 to integral submanifolds. There are however cosymplectic manifolds which are not globally the product of a Kahler manifold and a 1-dimensional manifold; a 3-dimensional example was given by Chinea, de Leon and Marrero [1993] and higher dimensional examples were given by Marrero and Padron [1998]. We have remarked that a Sasakian manifold is sometimes viewed as an odddimensional analogue of a Kahler manifold. In view of the present theorem the same can be said of cosymplectic manifolds. Moreover recalling the almost

complex structure J(X, f 1) = (OX - f , r7(X) A) on Men+l x R defined at

78

6. Sasakian and Cosymplectic Manifolds

the beginning of this chapter, consider the product metric G = g + dt2. An easy calculation shows that

G(J(X,f1d IJIYf2dtJl

G\\X,fid

(Yf4))

so that (M2n+1 x R, J, G) is an almost Hermitian manifold. Denote by V the Levi-Civita connection of G. If now one assumes that this structure is Kahler, then another straightforward computation gives

0 = (V(x,o)J)(Y,0) _

(Vx)(Y))

((vx

from which we see that M2,+1 is cosymplectic. Thus we have the question of the relation between the metric structure on Men+1 x R and the Sasakian condition. This was discussed by Tashiro [1963] and Oubina [1985]. Let G = e2pG' be a conformal change of a Kahler metric, then, as is well known, the respective connections are related by

DxY = DxY + (Xp)Y + (Yp)X - G'(X, Y)P where P = grad'p and we have

(DxJ)Y = (JYp)X - G'(X, JY)P - (Yp)JX +G'(X,Y)JP. Now suppose that the product metric G on Men+1 x II8 is conformally equiv-

alent to a Kahler metric G' with p = -t; then grad'p = -e-2tat. Thus for X, Y vector fields on Men+1

(D(x,o)J)(Y, 0) = -77(Y)(X, 0) + g(X, OY)

(0, dt) + g(X,

0)

on the one hand and on the other (D(x,o) J) (Y, 0) = D(x,o) (OY, rl (Y)

_

d) - J(VxY, 0)

(vx(x(Y))) - (OVXY,rl(OxY)dt)

.

Comparing the components, we see that (Vxql)Y = g(X, Y) - 77(Y)X and hence that Men+1 is Sasakian. Also (Vxi) (Y) = g(X, OY), since is now Killing. Conversely it is clear that if Men+1 is Sasakian, then (D(x,o) J) (Y, 0) is given

as above. Therefore making the inverse conformal change (D,(x,o) J) (Y, 0) = 0.

6.6 Products of almost contact manifolds

79

Moreover

(D(x,o) J)

(0, dt) = D(x,o)

0) = (-V

0) = (OX, 0)

and this is equal to

(DCx,o)J) (0d) -g(X,d+ (OX, 'q (X) Therefore (D'(x,o) J) (0, at) = 0 and similarly

(D(o.)J)(Y, 0) = 0,

(D(o dt)J) (0, dt) = 0.

The idea of using a deformed metric on M x R to characterize Sasakian manifolds can be done in another way which is taken as the definition of a Sasakian manifold in Boyer and Galicki [1999]. Let (M71, g) be a Riemannian manifold and ]l8+ the positive reals. Then (Mm, g) is Sasakian if and only if the holonomy group of (118+ x Mm, dr2+r2g) reduces to a subgroup of U(m21) Thus (R+ x Mm, dr2+r2g) is Kahler and m = 2n+1, n > 1. Boyer and Galicki are particularly interested in defining 3-Sasakian manifolds in an analogous way, see Chapter 13 and Boyer, Galicki and Mann [1994].

6.6

Products of almost contact manifolds

We continue our discussion in the last section with a look at products of almost contact manifolds. However we stress only the statement of results rather than

the details of their proofs; the interested reader may consult the references. Let M1 and M2 be almost contact manifolds with almost contact structures (c2, z, 71i), i = 1, 2. A. Morimoto [1963] defined an almost complex structure J on the product M1 x M2 by J(X1,X2) = (01X1 - 772(X2)S1, 02X2 + 771(X1)S2);

that J2 (X1' X2) = -(X1, X2) is an easy calculation. Morimoto then proved the following theorem.

Theorem 6.8 J is integrable if and only if both (01, i, r/i) and (02, S2, 772) are normal.

An interesting corollary of this is a result of Calabi and Eckmann [1953] that the product of two odd-dimensional spheres is a complex manifold. Corollary 6.6 S2P+1 x 529+1 is a complex manifold.

80

6. Sasakian and Cosymplectic Manifolds

Turning again to metric considerations, let M1 and M2 be almost contact metric manifolds with almost contact metric structures (0j, jj, riz, g;,), i = 1, 2. Capursi [1984] studied the product metric G = g1 + 92 and it is again an easy calculation that G(J(X1, X2), J(Yi, Y2)) = G((X1, X2), (Yl, Y2)).

The result of Capursi is the following.

Theorem 6.9 (M1 x M2, J, G) is Kahler if and only if both (Ml, 01, 1, i1, 91) and (M2, 02, 2, 772, 92) are cosymplectic.

In view of this and the observation of Tashiro and Oubina in the last section

on how to obtain the Sasakian condition from the structure on M x R, one might ask how to get both M1 and M2 Sasakian out of the almost Hermitian structure (J, G) on M1 x M2. To the author's knowledge this is an open question. On the other hand a special almost contact metric structure introduced by Kenmotsu [1972] seems to play a role here. An almost contact metric manifold (M, 77, g) is called a Kenmotsu manifold if it satisfies

(Vxc)Y = g(qX, Y) - rl(Y)gX. Kenmotsu [1972] gave a local characterization of this structure. Theorem 6.10 Every point of a Kenmotsu manifold has a neighborhood which is a warped product (-e, e) x f V where f (t) = cet and V is Kahler. Returning to the product space M1 x M2, define an almost complex structure by

J(X1, X2) _ (ql1X1 - e 2µT)2(X2)S1, 02X2 +

e-2,771

(XI)62)-

Again j2(X1, X2) = -(X1, X2) is an easy calculation. Let G = e2Pgl + e2Tg2 then G is Hermitian if and only if p = (p -T). Blair and Oubina [1990] then 2 noted the following.

Theorem 6.11 Let M1 and M2 be almost contact metric manifolds and U a coordinate neighborhood on M2 such that 62 = at. Consider the change of metric G = e2Pg1 + e2Tg2 on M1 x U where p = log(k - e-1) and rr = -t for some constant k. Then (M1 x U, J, G) is Kahler if and only if the structure on M1 is Sasakian and the structure on U is Kenmotsu.

A variation of the above result is that if (Ml x U, J, G) is Kahler for the conformal change p = T = -t, then M1 is Sasakian and the structure (02, -62, -772,92) is Kenmotsu.

The fact that this theorem is local in regard to the second manifold M2 is not unnatural. Even for M1 x R, the 1-dimensional case for M2, note that

6.7 Examples

81

the Hopf manifold S2n+1 x S' is locally conformally Kahler but not globally conformally Kahler. We close this section with a remark on a generalization of these structures. In the classification of Gray and Hervella [1980] of almost Hermitian manifolds

there appears a class, W4, of Hermitian manifolds which are closely related to locally conformally Kahler manifolds. Again consider M1 x Il8 with the almost complex structure J(X, f dt) = (OX and product metric G. Oubina [1985] introduced the notion of a trans-Sasakian structure as an almost contact metric structure (0, , 77, g) for which the almost Hermitian manifold (M1 x R, J, G) belongs to the class W4. This may be expressed by the condition

(VxO)Y = a(g(X, Y) - 77(1')X) +O(g(OX,Y) -'1(Y)cX) for functions a and,3 on M and the trans-Sasakian structure is said to be of type (a,,Q). If 0 but not a (respectively a but not 0) vanishes, the structure is a-Sasakian (resp. ,Q-Kenmotsu) (see also Janssens and Vanhecke [1981]). Marrero [1992] showed that a trans-Sasakian manifold of dimension > 5 is either a-Sasakian, ,Q-Kenmotsu or cosymplectic. He also showed that if M is a 3-dimensional Sasakian manifold with structure tensors (0, , 77, g), f a positive non-constant function and f g + (1 - f)77 ® 77, then (0, , 77, g) is trans-Sasakian of type (f, 20og f )).

6.7 Examples 6.7.1

R2n+1

In Example 4.5.1 we gave explicitly an associated almost contact metric structure g g) to the Darboux contact form 77 = (dz - > 1 yidxi) on From the matrix expression for 0 given in Example2 4.5.1 it is easy to check that R2n+1.

[0, 0] + 2d ® = 0 and hence that this contact metric structure is Sasakian. 6.7.2 Principal circle bundles

In Example 4.5.4 we saw that a compact regular contact manifold Men+1 carries a K-contact structure (0, , 77, g), defined in terms of the almost Kahler structure (J, G) of the base manifold Men. Since £ ql = N(3) = 0, [0, 0]

X) + 2dr7(e, X) = 02

X] -

OX] = 0.

Now OX = irJ7r,,X, so for projectable horizontal vector fields X and Y, 0] (X, Y) + 2dr7(X, Y) =

rJ27r., [X,

y] + [*J7r*X, rJ7r., Y]

82

6. Sasakian and Cosymplectic Manifolds

-rJ'ir*[X, FrJ7r*Y] + dr7(X, Y) -rJ2 [7r*X, rr*Y] + ir[J7r*X, J7r*Y] + r7 ([Fr J7r*X,,rJ7r*Y] )

-irJ[J7r*X, 7r*Y] - irJ[rr*X, J7r*Y] + dr7(X, Y) = i'r[J, J](7r*X,7r*Y) - 2(Sl(J7r,X, Jar*Y) 0 7r - 0(7r*X, rr*Y) o 7r) = ir[J, J](7r*X, ir,,Y).

Thus we see that the K-contact structure (0, , 77, g) is Sasakian if and only if the base manifold (M2n J, G) is Kahlerian (Hatakeyama [1963]). If (M2n J, G) is only almost Kahler, then (0, , 77, g) is only K-contact. Similar to the Boothby-Wang fibration of compact regular contact manifolds, A. Morimoto [1964] obtained a fibration of compact normal almost contact manifolds with 1; regular. First however, let rr : Men+1 _, M2n be a principal circle bundle over a complex manifold Men and suppose there exists a connection form q such that dr7 = 7r*W where' is a form of bidegree (1,1) on M2n. Then we again define OX = f?Jir,,X where J is the almost complex structure on M2n and -r the horizontal lift with respect to 77. Let t; be a vertical vector field with 77(1;) = 1. Then (0, E, 77) is an almost contact structure. Noting 0 by the definition of that ££97 = 0, since 77 is a connection form, and

0, a computation of Null similar to the one just given shows that Men+1 is a normal almost contact manifold (Morimoto [1963]). Conversely we give the following theorem of Morimoto [1964] and just sketch

its proof since the major ideas have already been given. Theorem 6.12 Let M2n+1 be a compact normal almost contact manifold with structure tensors (0, , 17) and suppose that E is a regular vector field. Then Men+1 is the bundle space of a principal circle bundle 7r M2,,+1 ) M2n over a complex manifold M2n. Moreover 77 is a connection form and the 2form I on M2n such that dr7 = rr* is of bidegree (1, 1). :

Proof. In the proof of the Boothby-Wang Theorem (Section 3.3) we defined the period function A of the vector field and showed that A was constant on M2n+1 which we then took to be 1. The argument (cf. Tanno [1965]) required only that 1 and £er7 = N(4) = 0. Thus we again have a circle bundle structure as in the Boothby-Wang fibration with q a connection form. Now since £eO = NN = 0, 0 is projectable and we can define an almost complex structure J on M2n by JX = 7r,,O rX where Fr denotes the horizontal lift with respect to 77. That j2 = -I is immediate and

6.7 Examples

83

,r[J, J] (X, Y) _ -[irX, irY] + i7([-rX, rY]) + [OrrX, c'rY] - y1([OFrX, O'TrY]) -0[0-rX, irY] - 0[-7rX, 0-7rY] _ [0, 0] (FrX, *Y) + 2dr)(qlirX,

[0, 0] (irX, FrY) + 2drl(-rX, irY) = 0,

the last equality following from N(' = 0 and the next to last from N(2) = 0. Finally T (JX, JY) orr = drl(ciirX, 07rY) = drl(-rX, irY) _ IF (X, Y) orr showing

that i is of bidegree (1, 1).

6.7.3 A non-normal almost contact structure on S5 In Example 4.5.3 we saw that S5 inherits from the almost Hermitian structure

on S6, an almost contact metric structure different than the standard one. Recall that the almost Hermitian structure (J, on S6 given in Example 4.5.3 is a nearly Kahler structure, i.e., (VxJ)X = 0 for all vector fields X. The geometric meaning of this condition is that geodesics are holomorphically planar curves. A curve ry on an almost Hermitian manifold is holomorphically

planar if the holomorphic section determined by its tangent field is parallel along the curve. We will now show that the induced almost contact metric structure (0, , r/, g) on S5 satisfies a similar condition, namely (VxO)X = 0. This is an immediate

consequence of the following theorem of the author [1971]; for notation see Example 4.5.2.

Theorem 6.13 Let Men+' be a hypersurface of a nearly Kahler manifold Men+2. Then the induced almost contact metric structure (0, , rl, g) satisfies (VxO)X = 0 if and only if the second fundamental form ci is proportional to (rl (D rl)v.

Proof. From equation (*) in Example 4.5.2 and OxY = VxY + c'(X, Y) we have

(Vx(D)(Y, Z) = (tx1)(Y, Z) + 9(ci(X, Y), v)77(Z) + 9(ci(X, Z), v)77(Y) where S2 is the fundamental 2-form of the nearly Kahler structure. Interchanging X and Z and adding we obtain (ox,D)(Y, Z) + (VzD) (Y, X) = -29(o,(X, Z), v)77(Y) +9(o(X, Y), v)r7(Z) + +§ (0'(Z, Y), v)77(X).

Now if o- is proportional to (rl®rl)v, (Vxql)X = 0. Conversely if (VxO)X = 0, 0 = -2g(ci(X, Z), v)i7(Y) + g(ci(X, y), v)rl(Z) + g(ci(Z, Y), v)77(X).

84

6. Sasakian and Cosymplectic Manifolds

Setting Y = gives 2g(o-(X, Z), v) = g(o (X, ), v)rl(Z)+g(o-(Z, ), v)rl(X) but v)77(Z) and consequently Z), v) now setting X = l we have . (o (X, Z), v) = v),7(X)i1(Z) A cosymplectic version of this theorem was given by Goldberg in [1968a] (see also Okumura [1965]). An almost contact metric structure (0, 6,,q, g) satisfying (VXc)X = 0 is called a nearly cosymplectic structure. As a further justification

of this name we remark without proof that a normal nearly cosymplectic manifold is cosymplectic (see the author's paper [1971] for details).

Proposition 6.1 On a nearly cosymplectic manifold

is a Killing vector

field.

= 0. Now differentiating the compatibility condition, g(OX, OY) = 9(X, Y) -rl(X)rl(Y), with respect to we find

Proof. Clearly (V 60)t; = 0 from which one easily obtains V

g((oO)X, OY) + g(qX, (V ql)Y) = 0. The nearly cosymplectic condition then gives g((V xo)e, OY) + g(qX, (VY0)e) = 0 which easily simplifies to

Y) +

X) = 0.

Proposition 6.2 On a normal nearly cosymplectic manifold, drl = 0. Proof. Since the structure is normal, N(' and N(2) vanish, thus setting Y = X and Z = in Lemma 6.1 we have drl(X, OX) = 0 for all vector fields X. Linearizing this we obtain OY) +d7(Y, OX) = 0, but from the vanishing of N(2), drl(X, qY) = -dri(OX, Y) and hence drl(X, cY) = 0. Also drl(X,

2 (g(Vx,', ) - g(Vgl;, X)) = 0 and so we obtain drl = 0. Turning now to S5 as a totally geodesic hypersurface of S6 (Example 4.5.3), its induced structure is nearly cosymplectic by Theorem 6.13. This structure is not normal, for if it were, then by our two propositions r) is respectively co-closed and closed and hence harmonic, contradicting the vanishing of the first Betti number of S5.

6.7.4

Men+l C M2n+2

We have already seen that the odd-dimensional spheres are Sasakian manifolds

and that this structure may be obtained both by viewing the sphere as a

6.7 Examples

85

) CPn hypersurf ace i n (Cn+1 or by con sid eri ng th e H opf fib ra tion 7r Stn+l as a special case of the Boothby-Wang fibration. In Theorem 4.12 we saw the condition for the induced almost contact metric structure on a hypersurface of a Kahler manifold to be contact metric. Now similar to Theorem 6.13 we give the condition for the hypersurface to be Sasakian; this is a result of Tashiro :

[1963] and we will omit the proof as it is similar to the proof of Theorem 6.13.

Theorem 6.14 Let Men+1 be a hypersurface of a Kahler manifold Men+z r], g) is Sasakian if Then the induced almost contact metric structure and only if the second fundamental form or = (-g + 0(r) ® rl))v, for some function ,3.

g) is said to be nearly Sasakian if (Vxc)Y+

A contact metric structure

(Vyq)X = 2g(X,Y) - ri(X)Y -,q(Y)X. The above theorem also holds for this structure on a hypersurface of a nearly Kahler manifold (Blair, Showers, and Yano [1976]). Similar to the structure on S5 obtained in Examples 4.5.3 and 6.7.3, consider S5 as an umbilical hypersurface of the unit sphere Ss at

a "latitude" of 45° so that o(X,Y) = -g(X,Y)v. Then the induced almost contact metric structure is nearly Sasakian but not Sasakian. For other results on hypersurfaces of Kahler manifolds see Okumura [1964a, 1964b, 1966], Vernon [1987].

6.7.5 Brieskorn manifolds

In this section we will show that the Brieskorn manifolds admit Sasakian structures which are often non-regular. Consider Cn+1 with coordinates z = (zo, ... , zn) and let (ao, ... , an) be an (n+1)-tuple of positive integers. A polynomial of the form P(z) = z0'°--. - -+za,n is called a Brieskorn polynomial and let Vzn(ao, ... , a,), or just V2n denote Ezn-1(ao, ... , a,) = V2n n 82n+1(1) the zero set of P. Then or just E2n-1 is l called a Brieskorn manifold. For w E C define fw : (Cn+1

Cn+1 by

rrzw

+n_w

fw(z) = (e a° zo, ... , e an zn)

where m is the least common multiple of (ao, ... , an). Gc = { fw l w E C} is an Abelian group of diffeomorphisms of Cn+l. Let Ga be the 1-parameter subgroup given by { f, I s E R}. The 118-action leaves V2n invariant and differen-

tiating with respect to s at s = 0, we see that the vector field which generates GR is

a=

m-za)

as

.

6. Sasakian and Cosymplectic Manifolds

86

Now let GS = { fitlt E R}; fit = fi(t+2,r) and is an Sl-action on Cn+1 leaving E2n-1 invariant. The vector field which generates GS is

m. b = is = (-tiza a«

The almost complex structure j on Cn+1 restricts to V2n and = -b = -Ja E2n-1. Moreover for an arbitrary vector field tangent is a vector field tangent to E2n-1, decomposing JX as to

JX = OX + rl(X)a gives

E2n-1,

an almost contact structure (0, , rl) as in Example 4.5.2; see

Sasaki and Takahashi [1976] and Sasaki [1985]. (In Sasaki [1985], the structure tensors have the opposite sign from our construction here.) Sasaki and Takahashi prove the following.

Theorem 6.15 The almost contact structure (0, , i) on the Brieskorn manifold

E2n-1 is normal.

Recalling the theorem of Morimoto, Theorem 6.8, we have the following result of Brieskorn and Van de Ven [1968].

Corollary 6.7 Let and

E2n-1 and E2m-1 be Brieskorn manifolds. Then E2n-1 X E2m-1 are complex manifolds.

E2n-1

xR

In regard to non-regularity, Sasaki and Takahashi prove the following.

Theorem 6.16 The almost contact structure (0, , rl) on the Brieskorn manE2n-1(ao, ifold . , an), is non-regular if and only if there exist three positive integers aa, aµ, a among the positive integers ao, ... , an such that the least common multiple of aa, a,,, is not equal to the least common multiple of aa, . .

a.,, a,,.

Now let H be the inner product of a with the position vector z c Cn+l, i.e.,

n zz, denoting the inner product on Cn+ by , H =< a, z >= m i:a=o a

Taking the inner product of JX = OX + i(X)a with the position vector z we Cn+1 denotes have H71(X) =< JX, z >= - < X, Jz >. Thus if t : E2n-1

the imbedding and w = 2 (zadza - zadza), 77 =

t

. 77 and t*w are contact a-o forms on E2n'1. This was shown by Sasaki and Hsu [1976] and simpler proofs given by Abe [1977] and Vaisman [1978]. Turning to the question of an associated metric for 77, set

g(X, Y) =

H(< X, Y > -77(X) < Y,

> -ri(Y) < X, >

6.8 Topology

87

+71(X)77(Y) < , >) +?7(X)'1(Y) > Cn+1 Note that g is not the induced metric from the imbedding i. E2n-1 E2n-1 Sasaki [1985] showed that (0, , 77, g) is a Sasakian structure on Thus we see that there are in fact many non-regular Sasakian manifolds. Much of the above goes over to more general spaces. For discussion of these generalizations, see the references mentioned as well as Abe [1976], Abe and Erbacher [1975], Lutz and Meckert [1976] and Sato [1977]. An earlier example of a non-regular Sasakian manifold was given by Tanno [1969]. :

6.8

Topology

In the 1960s a great deal of work was done on the topology of compact Sasakian and to a lesser extent cosymplectic manifolds. The idea was to see how much

a compact Sasakian manifold must be like a sphere. In the case of a compact Kahler manifold, the even-dimensional Betti numbers are different from zero and the odd-dimensional Betti numbers are even, properties which are certainly enjoyed by complex projective space (see e.g., Goldberg [1962, Chapter V]). Furthermore the Betti numbers by of a compact Kahler manifold of positive constant holomorphic curvature are equal to 1 for p even and vanish for p odd (see e.g., Goldberg [1962, Chapter VI]). More recently there has been renewed interest in the cosymplectic case. For brevity we will just mention a few results and the interested reader can consult the references for details. In [1965] Tachibana proved that the first Betti number of a compact Sasakian manifold Men+1 is zero or even. This is proved by first showing that on a compact K-contact manifold, a harmonic 1-form w is orthogonal to the contact form r7. Then letting w = w o 0 and computing the Laplacian of W one obtains the harmonicity of w as well. Thus the number of independent harmonic 1forms is even. The computation uses the fact that on a Sasakian manifold the Ricci operator commutes with 0, a fact that we will prove in our subsequent discussions of curvature. More generally the p-th Betti number is even for p

odd and 1 < p < n and by duality for p even and n + 1 < p < 2n (Fujitani [1966], see also Blair and Goldberg [1967]).

Considerable attention has been given to the vanishing of the second Betti number under some curvature restrictions as well as being isometric to the unit sphere under stronger conditions. A compact Sasakian manifold of strictly positive curvature has vanishing second Betti number (Moskal [1966], Tachibana

and Ogawa [1966], Tanno [1968], Goldberg [1967] in the regular case and Goldberg [1968b] in the regular case with non-negative curvature). A compact, simply-connected Sasakian-Einstein space of strictly positive curvature

6. Sasakian and Cosymplectic Manifolds

88

is isometric to the unit sphere (Moskal [1966]). Pinching theorems have been obtained by Tanno [1968] including an analogue of holomorphic pinching. More recently Goldberg [1986] showed that a compact simply-connected regular Sasakian manifold M of strictly positive curvature is homeomorphic to a sphere. If, in addition, M has constant scalar curvature, then Goldberg had shown earlier [1967] that M is isometric to a sphere, but not necessarily with a constant curvature metric (cf. the metrics on the sphere in Example 7.4.1).

Allowing some negative curvature, Tanno [1968] showed that if M2n+1 is a compact K-contact manifold with sectional curvature greater than 2n31, then b1 = 0. Similarly if the Ricci tensor p is such that p + 2g is positive definite, then b1 = 0. By duality in dimension 3, one also has b2 = 0.

In dimension 5, Perrone [1989] showed that if M5 is a compact simplyconnected regular Sasakian manifold with b2 = 0 and with scalar curvature T > -4, then M5 is homeomorphic to a sphere. If, in addition, M5 has constant scalar curvature, M is isometric to a sphere (but not necessarily with a constant curvature metric). A classical result on the topology of a compact Kahler manifold Men is the monotonicity of the Betti numbers, namely by < by+2, p < n -1; this is proved using the idea of effective harmonic forms (see Goldberg [1962, Chapter V]). We briefly describe the idea of effective forms in the almost contact context. Let (M2n+1 g) be a compact almost contact manifold and, as before, we denote the fundamental 2-form by (D. Define two operators L and A acting on differential forms a by

La =an(b, Aa=*L*a where * denotes the Hodge star isomorphism (La = a A Q in the symplectic case). A p-form a is said to be effective if Aa = 0. We now have the following Proposition (see Blair and Goldberg [1967], Chinea, de Leon and Marrero [1993] ).

Proposition 6.3 On an almost contact metric manifold of dimension 2n+1, every p-form a with p < n + 1 may be written uniquely as a sum

a=

Lk p-2k k=0

where the 73p-2k's, 0 < k < r, are effective forms of degree p - 2k and r = IV.

In the cosymplectic case we cite the recent work of Chinea, de Leon and Marrero [1993]. They prove that on a compact cosymplectic manifold M2'n+1 bo < b1 < < bn = bn+1 and bn+1 >- bn+2 > .. >- b2n+1. Moreover the

6.8 Topology

89

differences b2p+1 - b2p with 0 < p < n are even and so in particular the first Betti number of Men+1 is odd. They also prove a strong Lefschetz property for a compact cosymplectic manifold and show that such a manifold is formal (i.e., the homotopy type of the differential graded algebra of differential forms is the same as the homotopy type of the cohomology ring). This topological work involves generalizing from Kahler geometry notions of the bidegree of differential forms, effective harmonic forms, etc. For studies of the tridegree of differential forms and applications, see Chinea, de Leon and Marrero [1997], Moskal [1977] and Fujitani [1966]. A p-form a is said to be coeffective if La = 0. Coeffective cohomology was studied in the symplectic case by Bouche [1990] and in the almost cosymplectic case by Chinea, de Leon and Marrero [1995]. In both papers the relation between the coeffective and the de Rham cohomologies of the manifolds is discussed (for the almost contact case see also M. Fernandez, R. Ibanez and M. de Leon [1997], [1998]).

7 Curvature of Contact Metric Manifolds

7.1

Basic curvature properties

In this chapter we discuss many aspects of the curvature of contact metric manifolds. We begin with some preliminaries concerning the tensor field h. Let Men+1 be a contact metric manifold with structure tensors rl, g) and h =

a £g¢ as before. Recall that in Lemma 6.2 we saw that Vx = -OX - QbhX. Proposition 7.1 On a contact metric manifold M2n-}-1 we have the formulas

(Vgh)X = OX - h2OX - ORxg 2 (Rgx6

-

,

h2X + 02X.

Proof. We compute Rgx = V Vx6-VxVgg-V[g,x]6 using Vxe = -OXqlhX ; thus

Rgxe = Vg(-OX - cbhX) +

X] +

X].

Applying ¢ and recalling that V = 0 we have

gRgx = vg(X + hX) - i(og(X + hX)) -

X] +

X]) -

= (Ogh)X + Vx + Using Vx = -OX - chX and Oh + ho = 0 (Lemma 6.2) this becomes ebRgxe = (Vgh)X - OX + h20X

X]

92

7. Curvature of Contact Metric Manifolds

which is the first formula. Now from the first formula we have

R£xe = h2X + 02X - 0(0£h)X and

ORS,6xe = -h2X - 02X

-

subtracting then yields the second formula.

Corollary 7.1 On a contact metric manifold Men,+1 the Ricci curvature in the direction e is given by

Ric(e) = 2n - trh2. Proof. Choosing X to be a unit vector orthogonal toe, the inner product of X with the second formula yields the following formula for sectional curvatures

K(e, X) + K(e, OX) = 2(1 - g(h2X, X)).

Therefore if {X,,... , X,, OX1, ... , OX,,., e} is a 0-basis, then summing over {X,,... , X,,} yields the result. Recall that a K-contact structure is a contact metric structure for which the vector field e is Killing and that this is the case if and only if the symmetric operator h vanishes. Thus from the above corollary we have the following immediate result (Blair [1977]).

Theorem 7.1 A contact metric manifold M2"+1 is K-contact if and only if Ric(e) = 2n.

With regard to sectional curvature we have an earlier result obtained by Hatakeyama, Ogawa and Tanno [1963].

Theorem 7.2 A contact metric manifold is K-contact if and only if the sectional curvature of all plane sections containing e are equal to 1. Moreover, on a K-contact manifold,

Proof. In view of the above the sufficiency is clear. Conversely if the structure

is K-contact, then since Vxe = -OX, we have for X orthogonal to

Rexe = V (-OX) + 0[e, X] = -OVxe = O2X = -X. The second statement is easily obtained.

Furthermore for the Ricci operator Q acting one we have the following.

7.1 Basic curvature properties

93

Proposition 7.2 On a K-contact metric manifold Men+1 Q = 2nd. Proof. Since l is Killing, it is affine and therefore

VxVY - Vvxy = From this we have (Vxcl)Y = RxeY but in Section 6.1 we saw that on a contact metric manifold Vio'j = -2n17 . Thus letting {XA} be a local orthonormal basis of the contact subbundle,

Q _) , R£xAXA = > ,(VxAO)XA = 2nd. We noted in Theorem 7.2 that on a K-contact manifold Rx£ = X -q(X)l;. On a Sasakian manifold we have the following stronger result.

Proposition 7.3 On a Sasakian manifold Rxy = 71(Y)X - 77(X)Y.

Proof.

Rxy = -VxOY+VygX + O[X,Y] = -(Vxo)Y + (Vyo)X = rl(Y)X -,q(x)y. As converses of these results one often sees propositions of the following type (Hatakeyama, Ogawa and Tanno [1963]).

Proposition 7.4 Let (M2ri+1, g) be a Riemannian manifold admitting a unit Killing field such that X for X orthogonal to . Then M2,+1 is a K-contact manifold.

Proof. Let rl(X) = g(X, l;) and OX = -Vxl;. Since is unit Killing V = 0 and

VxVyS - Vv.y = RxeY.

(*)

Thus for X orthogonal to O2X

= DoX =

and O = 0. Therefore 02 = -I + drl(X,Y) = 2-(g(Vx6,Y) -

17

-X

. Moreover X) = g(X, OY).

94

7. Curvature of Contact Metric Manifolds

n, g) is a contact metric structure on M2,+1_

Therefore

An interesting variation is a result of Rukimbira [1995b] that if a Riemannian manifold admits a unit Killing field such that the sectional curvatures

of plane sections containing l; is positive, then it also admits a K-contact structure but with a possibly different metric.

Proposition 7.5 If in Proposition 7.4 Rxy =

Y)X - g(X, e)Y, then

Men+1 is Sasakian.

Proof. From equation (*) in the proof of Proposition 7.4, (Vxq)Y = RexY and hence g((Vxq5)Y, Z) =

Z) = g(Ryz X) = g(g(Z)Y - il(Y)Z,X) ,

0

These results start with a Riemannian structure and construct the desired structure. However given a contact metric structure we have the following proposition.

Proposition 7.6 A contact metric structure is Sasakian if and only if

Rxy = il(Y)X - q(X)Y. Proof. For X orthogonal to 1;, Rgx = -X and so from the second equation of Proposition 1 we have

2 (-X - 0(-OX)) = h2X + O2X = h2X - X.

Therefore h2 = 0, but h is a symmetric operator and so h = 0. Thus

is

Killing and the result follows as above.

Finally for future use we establish some additional lemmas on Sasakian manifolds. The reader will recognize these curvature properties as being analogous to well-known curvature properties of Khhler manifolds. Let Men+1 be a Sasakian manifold with structure tensors rl, g) and define a tensor field P of type (0,4) by P(X, Y, Z, W) = drl(X, Z)g(Y, W) - drl(X, W)g(Y, Z)

-dil(Y, Z)g(X, W) + drl(Y, W)g(X, Z).

Lemma 7.1 On a Sasakian manifold we have a) g(RxyZ, OW) + g(RxygZ, W) = -P(X, Y, Z, W).

7.2 Curvature of contact metric manifolds

For X, Y, Z, W orthogonal to

95

we have

b) g(Rox OyOZ, OW) = g(RxyZ,W)

and c) g(RxoxY, OY) = g(RxyX, Y) + g(RxoyX, OY) - 2P(X, Y, X, OY).

Proof. A direct computation or the Ricci identity shows that

(VxVyI) -VyVx-D

-g(RxyZ,OW) -g(RxyOZ,W)

Computing the left-hand side using (V )Y = g(X, Y) -,q(Y)X yields a). Using a) and the definition of P we obtain b). Finally applying the first Bianchi identity to g(Rx,xY, qY) and using a) we obtain c).

Lemma 7.2 On a Sasakian manifold Qq = qQ. Proof. Choosing a 0-basis {Xi, Xn,+i = OXi, 6} we have for X and Y orthog-

onal to , g(QOX, OY) = L g(R,,x XAXA, OY) + g(Rox

OY)

A=1 zn

E g(Rox ox, OXA, OY) + g(X, Y) A=1

= g(QX,Y) where we have used b) from the previous lemma. We already know that Q _ 2ne and hence the Ricci operator Q commutes with 0 on a Sasakian manifold.

7.2

Curvature of contact metric manifolds

Before giving our main curvature results, we present some rather complicated lemmas from the paper [1979] of Olszak.

Lemma 7.3 On a contact metric manifold (V xc&)Y + (V,kx0)0Y = 2g(X, Y) - i7(Y) (X + hX + r, (X))

96

7. Curvature of Contact Metric Manifolds

Proof. Using Corollary 6.1 or by direct differentiation of VY = -OY-OhY we obtain (V x-1)) (OY, Z) - (V x b) (Y, OZ) = -rl (Y)g(X +hX, OZ) - rl (Z)g(X +hX, OY) (*)

and replacing Z by OZ and using Corollary 6.1 again we obtain

Z) = rl(Y)g(X + hX, Z) - rl(Z)g(X + hX,Y).

(Vx(D)(0Y, OZ) +

(**)

Now since dD = 0 we have

(VxID)(Y, Z) + (VY4))(Z,X) + (Vz(D)(X,Y) +(omx ID) (OY, Z) + (V Y II) (Z, OX) + (oz'D) (cX, OY)

+(VOx41)(Y,OZ) + (VYD)(OZ,OX) + (V z')(OX,Y)

-(ox(D)(cY,OZ) - (V ')('Z,X) - (VOA) (X, OY) = 0. Now (*) and (**) give (V xd))(Z, OY) +

Y)

= 2i7(Z)g(X,Y) - r,(Y)g(X + hX, Z) - ri(X)rt(Y)ri(Z) from which the result follows.

Lemma 7.4 The curvature tensor of a contact metric manifold satisfies

Z) = -(Vxd))(Y, Z) - g(X, (VyOh)Z) + g(X, (Vzoh)Y), g(R6xY, Z) OZ) + OZ) + g(R6 axOY, Z) = 2(VhA)(Y, Z) - 2ri(Y)g(X + hX, Z) + 2r7(Z)g(X + hX, Y).

Proof. Differentiating Vz = -OZ - OhZ we have Ryze _ -(VYO)Z + (V zo)Y - (DYq5h)Z + (V zOh)Y which, since d (D = 0, yields the first formula. Now set

A(X, Y, Z) _ - (VA) (Y, Z) + (V x11) (OY, OZ) -(V x,D) (Y, OZ) + (V mxID) (OY, Z) and

B(X, Y, Z) = -g(X, (Vy h)Z) + g(X, (V yoh)gZ) -g(OX, (VYOh)OZ) - g(OX, (V ych)Z).

7.2 Curvature of contact metric manifolds

97

Then by the first formula, the left side of the second formula is A(X, Y, Z) + B(X,Y, Z) - B(X, Z, Y). Now by Lemma 7.3 and equation (**) in its proof we have

A(X, Y, Z) = 2g(X, Y)77(Z) - 2g(X, Z)77(Y).

Also it is straightforward to show that r7((V yh)Z) = g(-Y + hY, hZ). Now rewrite B as

B(X,Y, Z) = -g(X, (Vyq)hZ) + g(X, h(VyO)Z) +g(X, hql(VLYO)Z) + g(X, qS(V yO)hZ) + i7(X)77((V yh)Z).

Then using Lemma 7.3 again

B(X, Y, Z) = 2g(hX, (Vyo)Z) + 2r7(Z)g(hX, Y) - 2r7(X)g(hY, hZ).

Finally computing A(X, Y, Z) + B(X, Y, Z) - B(X, Z, Y) and using d4) = 0 we have the result.

Let p denote the Ricci tensor and rr the scalar curvature. In addition we define the *-Ricci tensor p* and *-scalar curvature T* by contracting the curvature tensor by 0 instead of the metric. Precisely

*_

pij =

Riklm

oklojm'

T* = p*ii

These notions have their origin in almost Hermitian geometry and we shall see their almost Hermitian analogues in Section 10.2. We now have an important proposition due to Olszak [1979].

Proposition 7.7 On a contact metric manifold Men+1, T* - T + 4n2 = trh2 + 2 (IVOi2

- 4n) > 0

with equality if and only if Men+1 is Sasakian.

Proof. We have seen that DiOij = -2nr7j and Vke _ -O'k - O'mhmk Therefore using 02 = -I + r) ® and basic properties of h (Section 6.4) we have

OkjVkViOi = -4n2.

(*)

Differentiation of 02 = -1+ 77 ® yields clkjVtc5kj = 0. Since dD = 0, we have Ok'VkOtj = Ok'(-VtOik - V kt) from which we obtain Okivkctj = 0. In turn

OkjV,VkOtj = -(Vtck')(Vkcbtj) Using d1 = 0 on the second factor on the right and simplifying we have

Ok'VtOkOtj = -2ID0I2.

(**).

98

7. Curvature of Contact Metric Manifolds

From (*) and (**) Okj(VkVtOtj - VtVkOtj) = 210012 - 4n2. Therefore by Corollary 7.1 2Io0I2 - 4n2 = 0kj(Rktat0aj - RktjaOta) =

(9ka

-

+T*

=T*-T+(2n-trh2) which is the desired formula. gikcj + 0 is equivalent to Now (ViOjk - 9ikTlj + IVOI2-4n > 0 giving the inequality and by Theorem 6.3 equality holds if and only if the structure is Sasakian. We now prove the following important result of Olszak [1979].

Theorem 7.3 If a contact metric manifold M2"+i is of constant curvature c and dimension > 5, then c = 1 and the structure is Sasakian. h2X +02X; thus if Rx yZ = c(g(Y, Z)X - g(X, Z)Y), then 2 (77(X)e - X + cl2X) = h2X +

Proof. Recall from Proposition 7.1 that, 1(RCx -

ql2X. Therefore h2X + (c - 1)02X and hence trh2 = 2n(l - c). Now from Lemma 7.4

c(9(X,Y)?7(Z) - J1(Y)9(X, Z)) - 0 - crl(Y)9(gX, OZ) + cr7(Z)9(OX, cY)

= 2(VhxD)(Y, Z) - 277(Y)g(X + hX, Z) + 277(Z)g(X + hX, Y). Therefore

(Vhxd))(Y, Z) = (1 - c)(r7(Y)9(X, Z) - r)(Z)9(X,Y)) +r7(Y)g(hX, Z) - 77(Z)g(hX,Y).

Replacing X by hX, we have

-9((V(C-l)(-x+,n(x)&)Y, Z) = (1 - c)(rl(Y)9(hX, Z) - 77(Z)9(hX,Y))

Z) - i (Z)g((c - 1)(-X +

+r7(Y)g((c-1)(-X and hence

(Vxql)Y = g(X + hX,Y) - r7(Y)(X + hX). Using this to compute Ip(hI2 and using trh2 = 2n(1- c) we obtain 1

IV

2 = 4n(2 - c).

7.2 Curvature of contact metric manifolds

99

On the other hand T = 2n(2n + 1)c and r* = 2nc as is easily checked. Now from Olszak's formula in Proposition 7.7,

T - T* - 4n2 = -trh2 - 2 IVOl2 + 2n < 0 with equality if and only if the structure is Sasakian. Computing both sides of this formula we find that 4n2(c - 1) = 4n(c - 1); thus if n > 1, c = 1 and hence r - T* - 4n2 = 0 giving Men+1 Sasakian. Earlier the present author [1976] showed that in dimension > 5, there are no flat associated metrics. Thus while the 5-torus carries a contact structure (Example 3.2.7), the flat metric is not an associated metric. In dimension 3, the only constant curvature cases are constant curvature 0 and 1 as we will note below. The non-existence of flat associated metrics does raise the question as to whether there are contact metric manifolds of everywhere non-positive curvature, except for the flat 3-dimensional case. If the manifold is compact and we ask for strictly negative curvature, we can answer this question in the negative using the following deep result of A. Zeghib [1995] on geodesic plane fields. Recall that a plane field on a Riemannian manifold is said to be geodesic if any geodesic tangent to the plane field at some point is everywhere tangent

to it. Theorem 7.4 A compact negatively curved Riemannian manifold has no C1 geodesic plane field (of non-trivial dimension). Since for any contact metric structure the integral curves of 1; are geodesics (Theorem 4.5), E determines a geodesic line field to which we can apply the theorem of Zeghib as was pointed out by Rukimbira [1998]. Thus we have the following corollary.

Corollary 7.2 On a compact contact manifold, there is no associated metric of strictly negative curvature.

The author conjectures that this and a bit more is true locally, viz., that except for the flat 3-dimensional case, any contact metric manifold has some positive sectional curvature. The fact that hyperbolic space has many 1-dimensional totally geodesic foliations does not violate such a conjecture, since by the above theorem of Olszak the hyperbolic metric cannot be an associated metric of any contact structure. Along the line of the influence of a contact structure on the curvature of its associated metrics we also make the following remark. In [1941] Myers proved

that a complete Riemannian manifold for which Ric > 6 > 0 is compact. In [1981] Hasegawa and Seino generalized Myers' theorem for a K-contact manifold by proving that a complete K-contact manifold for which Ric >

100

7. Curvature of Contact Metric Manifolds

-8 > -2 is compact (they state their result in the Sasakian case, but their proof uses only the K-contact property). As we have seen in the K-contact case, all sectional curvatures of plane sections containing

are equal to 1 and

hence there is a lot of positive curvature from the outset. In an attempt to weaken the K-contact requirement in this result, Blair and Sharma [1990a} considered a contact metric manifold Men+i for which is an eigenvector field of the Ricci operator, equivalently divho is proportional to 77. In this case if

Ric > -5 > -2 and the sectional curvatures of plane sections containing

are >e>6' >Owhere

5'=2\/n(5-2 25+n+2)-(6-2 25+1+2n), then Men+1 is compact. The flat case was investigated further by Rukimbira [1998] who showed that

a compact flat contact metric manifold is isometric to the quotient for a flat 3-torus by a finite cyclic group of isometries of order 1,2,3,4 or 6. In Example 3.2.6 and Section 6.2 we saw explicitly a flat contact metric structure on R3 and in turn on the 3-dimensional torus T3. As in Section 6.2 let y = (cos x3dx'+sin x3dx2) be the contact form; gig = 1 is an associated 2 the non-zero eigenvalues of h were ±1. It is interesting to study this metric and example in Darboux coordinates and to generalize it to higher dimensions. By the Darboux Theorem there exist coordinates (x, y, z) such that the contact form y is given by (dz - ydx). Consider the map f : R' --+ R' given by 2

xr = z cos x - y sin x,

x2 = -z sin x - y cos x,

x3 = -x.

Then (dz - ydx) = f *y and go = f *g is a flat associated metric for the z form 770 = 2 (dz - ydx). The metric go is given by Darboux 1

go =

4

1+y2+z2 z -y z

1

0

-y

0

1

Now consider the Darboux form y = (dz - > yidxi) on a 1

g=

4

Sij + viyj + 6ijz2

6Z'jz

- yi

Sip z

5i.7

0

-y3

0

1

R2n+1.

is an associated metric quite different from the Sasakian metric given in Examples 4.5.1 and 6.7.1. Direct computation shows that hay; Therefore -1 is an eigenvalue of h of multiplicity n and hence in turn +1 is also an eigenvalue of multiplicity n. This metric enjoys the following curvature properties:

7.2 Curvature of contact metric manifolds

R£x = 0 for every X and Rxye = 0 for X, Y E {+1}. However Rxy

101

0

in general, for example in dimension 5, 1

Ra a0

2(-y

2a ax1

1a

2

a

1Z

+y axe +y zayl -y ay2

We now show that the condition Rxyt; = 0 for all X, Y has a strong and interesting implication for a contact metric manifold, namely that it is locally the product of Euclidean space E11+1 and a sphere of constant curvature +4 (Blair [1977]).

Theorem 7.5 A contact metric manifold M2n+1 satisfying RxYe = 0 is locally isometric to En+1 X Sn(4) for n > 1 and flat for n = 1.

Proof. Rx yI = 0 implies, by Proposition 7.1, that h2 + l2 = 0 and hence that the non-zero eigenvalues of h are ±1, each with multiplicity n. For X, Y E {- 1}, 0 = - V [x,Yl = O [X Y] + q5h[X, Y] ,

and i([X,Y]) _ -2dg(X,Y) _ -2g(X,OY) = 0. Thus {-1} is integrable. Also 0 = so that [X,1;] E {-1}. Therefore {-1} is integrable. Choose coordinates (u°, ... , u2n) such that aao ... aan E {-1} ® Let Xi = fi aa, where the fi 's are functions chosen so that X. E o ,

E

{+1}. Now [ate Xi] E {-1}{}, k= 0,... n and hence

V[_ 8 xtilc_V V e-VxV =-2V KOXi au

au

au

from which V Oxi oxi = 0.

We therefore see that the integral submanifolds of {-1} ®

are totally

geodesic and flat.

From the second formula of Lemma 7.4, we have for X, Y, Z orthogonal to that g((Vxo)Y, Z) = 0. Also for X, Y E {+11,

0 = Rxy _ -2Vx4Y + 2VygX - V [x,Y] _ -2(Vxq)Y + 2(Vyq )X - 20[X, Y] + O[X, Y] + Oh[X, Y].

Taking the inner product with Z E {+1}, gives g(-O[X, Y] - hq[X, Y], Z) _

0 and hence g(O[X, Y], Z) = 0. Thus [X, Y] is orthogonal to {-1}. Also ri([X,Y]) = 0 and therefore {+1} is integrable. Now take X E {-1} and Y E {+1}; then

0 = Rx y = -2V Y + O[X, Y] + ch[X, Y]

102

7. Curvature of Contact Metric Manifolds

= -2(Vxql)Y - gVxY - glVYX - hOVxY + hOVyX. Taking the inner product with Z E {-1}, we have g(¢VyX, Z) = 0 and hence

that VyX is orthogonal to {+1}. Also Vy = -20Y is orthogonal to {+1}. Therefore the integral submanifolds of {+1} are totally geodesic. Moreover we are now at the point that Men+i has a local Riemannian product structure. For X E {+1},

g((Vxgl)Y, 6) = -g((Vxq)6, Y) = g(OVx6, Y) = g(O(-20X ), Y) = 2g(X, Y)

and hence by the second formula of Lemma 7.4, (VxgS)Y = 2g(X,Y) for X, Y E {+1}. Now for X, Y, Z, W E {+1},

g(VxVYOZ, OW) - g(VxVYZ, W)

= g(Vx(2g(Y, Z) + OVyZ), OW) - g(VxVyZ, W) = -4g(Y, Z)g(X, W) using the property we noted above that g((VxO)Y, Z) = 0 for X, Y, Z orthogonal to l;. Using this property again to treat the bracket terms we have

g(Rx yOZ, OW) - g(RxyZ, W) _ -4(g(Y, Z)g(X, W) - g(X, Z)g(Y, W)),

but g(RxycZ,cW) = 0 by virtue of the Riemannian product structure. Therefore the integral submanifolds of {+1} are locally isometric to S' (4). In dimension 3, the integrability of {-1} and {+1} is immediate and the rest of the proof goes through giving that M3 is flat.

As a generalization of both Rxy = 0 and the Sasakian case, Rxye _ rt(Y)X - 77(X)Y, consider

Rxye = r,(r1(Y)X - rl(X)Y) + u(,7(Y)hX - 97(X)hY) for constants ic and A. We say this is a contact metric structure for which l; belongs to the (rc, a)-nullity distribution or we refer to this condition as the (rc, µ)-nullity condition or the contact metric manifold as a (rc, µ)-manifold. Despite the technical appearance of this condition there are good reasons for considering it; we first mention them here referring to Blair, Koufogiorgos, and Papantoniou [1995] for details and then give a classification theorem due to Boeckx [2000].

Theorem 7.6 A contact metric manifold M for which belongs to the (rc, A) nullity distribution is a strongly pseudo-convex CR-manifold.

7.2 Curvature of contact metric manifolds

103

Theorem 7.7 Let M2,+1 be a contact metric manifold for which belongs to the (rc, µ)-nullity distribution. Then n < 1. If i = 1, the structure is Sasakian. If r, < 1, the (rc, µ)-nullity condition determines the curvature of M27+1 completely.

We remark that when rc = 1, the proof shows that h = 0 and hence µ is indeterminate in this case. When rc < 1, the non-zero eigenvalues of h are 4- V 'I- rc each with multiplicity n. Let A be the positive eigenvalue. Explicit formulas for the curvature tensor may be found in Blair, Koufogiorgos, and Papantoniou [1995]. We mention only that for a unit vector X E [.\], ¢X E [-A] and we have

K(X, ) = n + aµ, K(OX, ) _ n - Aµ,

K(X, qX) = -(rc + µ).

Thus, turning to the sign of the curvature question, if a (rc, µ)-manifold were of negative curvature, then A > 1 by Corollary 7.1 and rc ± Aµ < 0, giving A2 - 1 > Alµl. K(X, ¢X) _ -(rc + µ) < 0 would then give Ajµj < a2 -1 < µ, a contradiction.

Theorem 7.8 Let M be a 3-dimensional contact metric manifold for which belongs to the (rc, p) -nullity distribution. Then M is either Sasakian or locally isometric to one of the unimodular Lie groups SU(2), SL(2, R). E(2), E(1, 1) with a left invariant metric.

Theorem 7.9 The standard contact metric structure on the tangent sphere bundle T1M (see Section 9.2) satisfies the (rc, ,u)-nullity condition if and only if

the base manifold M is of constant curvature. In particular if M has constant curvature c, then rc = c(2 - c) and µ = -2c.

Finally given a contact metric structure structure =ark,

=

rl, g), consider the deformed

1

ae, _0, 9=ag+a(a-1)rt®,

where a is a positive constant. Such a deformation is called a D-homothetic deformation, since the metrics restricted to the contact subbundle D are homothetic. This deformation was introduced by Tanno in [1968] and has many applications. While such a change preserves the state of being contact metric, K-contact, Sasakian or strongly pseudo-convex CR, it destroys a condition

like Rxy = 0 or Rxye = rc(rl(Y)X - 77(X)Y). However the form of the (rc, p)-nullity condition is preserved under a D-homothetic deformation with K

rc+a2-1 a2

'

µ=

+2a-2 a

7.

104

Curvature of Contact Metric Manifolds

Given a non-Sasakian (rc, A)-manifold M, Boeckx [2000] introduced an invariant

1-Y

IM =

V1-ti

and showed that for two non-Sasakian (it, µ)-manifolds (Mi, 0i, i, 7?i, gi), i = 1, 2, we have IM, = IM2 if and only if up to a D-homothetic deformation, the two spaces are locally isometric as contact metric manifolds. Thus we know all non-Sasakian (i, µ)-manifolds locally as soon as we have for every odd dimension 2n + 1 and for every possible value of the invariant I, one (i,,1)manifold (M, 0, , 71, g) with IM = I. From Theorem 7.9 we see that for the standard contact metric structure on the tangent sphere bundle of a manifold of constant curvature c, I = . Therefore as c varies over the reals, I takes on every value > -1. Boeckx now gives an example for any odd dimension and value of I < -1; his construction is as follows. Let g be a (2n+ 1)-dimensional Lie algebra. Introduce a basis for g, X1, X, Y1, ... , Y,t}, and for real numbers a and 0 define the Lie bracket by 1

Xl]

[

,

I'i]=

X2 - 21'1, 22X1

- aY2, 2

ic

X2] =

X, -

2] _

2

2

[X1,Xi] = aXi, i l 1, [Y2, Y] = l3Y, i

2,

[X1, Y1] = -/3X2 + [X2, Y1] = (3X1 - aY2,

-Y2,

X2 + PY, [Xi,Xj]

Y] _

= 0, i,j

2Y,

2

i > 3,

XZ, i > 3,

1,

[Y, Y] = 0, i, j 0 2, [X i, Y] = 0, i > 2,

[X2, Y2] = aY1 +

[Xi, Y1] = -aY, i > 3,

Xi] _

[X2, Y] = 8Xi, i > 3,

[Xi, Y] = 0, i > 3,

[Xi, l ' ] = S (-,3X2 + oY1 + 2e), i, j > 3. .;

The associated Lie group G is not unimodular for dim g > 5 and not both a and ,6 equal to zero, since tr adX, = (n-1)a and tr ady2 = (n-1),i3. Now define a metric on G by left translation of the basis X1 i ... , X,,,, Y1, ... , Y,}, taken as orthogonal at the identity. Then taking 17 as the metric dual of and defining

0 by O = 0, OXi = Y. and qlY = -Xi, we have a contact metric structure on G. Now for the present purpose suppose that 02 > a2. G is a non-Sasakian (i, µ)-manifold and

IG=-02-+a a > 0, IG attains any value < -1.

7.2 Curvature of contact metric manifolds

105

Returning to Theorem 7.8, consider the Lie algebra a2

X) _ - 2 Y,

[, Y] =

N-X,

[X, YJ = 2

which corresponds to a unimodular Lie group. Boeckx [2000] points out that

for appropriate values of a and 0 we obtain left invariant contact metric structures with values of the invariant IG as follows: Isu(2) > 1, IE(2) = 1, -1 < ISL(2,R) < 1, IE(1,1) = -1 and IsL(2,a) < -1 (see also Blair, Koufogiorgos, and Papantoniou [1995]). Finally in [2000] Koufogiorgos and Tsichlias considered the question of contact metric manifolds for which belongs to the (ic, µ)-nullity distribution but rc and it are functions rather than constants. They showed that in dimensions

> 5, i and y must be constant and in dimension 3 gave an example where rc and p are not constants; this case is studied further in Koufogiorgos and Tsichlias [toap].

If on a (r,, p)-manifold, y = 0, the contact metric manifold is said to be one for which 6 belongs to the ic-nullity distribution. In general the rc-nullity distribution of a Riemannian manifold (M, g) is the subbundle N(k) defined by

Np(rc) = {Z E TpMlRXYZ = rc((g(Y, Z)X - g(X, Z)Y) V X,Y E T,M}.

In dimension 3, Blair, Koufogiorgos, and Sharma [1990] showed that belonging to N(ic) is equivalent to the Ricci operator Q commuting with c and equivalent to the contact metric manifold being 77-Einstein, i.e.,

Q=al+brl®6 for some functions a and b. In dimensions > 5 it is known that for any Einstein K-contact manifold, a and b are constants. The main result of the

77-

author's papers with Koufogiorgos and Sharma [1990] and with H. Chen [1992]

is that a 3-dimensional contact metric manifold for which 6 belongs to the K-nullity distribution is either Sasakian, flat or locally isometric to a leftinvariant metric on the Lie groups SU(2) or SL(2, R). An interesting side question associated with this case is the dimension of the n-nullity distribution itself. Clearly if a vector Z belongs to N(rc), then the sectional curvatures of all plane sections containing Z are equal to ic. In particular on any Riemannian manifold, N(rc) is non-trivial for at most one value of rc. It is known that N(i) is an integrable subbundle with totally geodesic leaves of constant curvature ic, see e.g., Tanno [1978a]. In unpublished work, Baikoussis, Koufogiorgos and the author showed that if the characteristic vector field on a contact metric manifold M2'"+1, n > 1 belongs to N(rc), then

106

7. Curvature of Contact Metric Manifolds

(,c < 1 and) if ,c < 1 and is # 0, dimN(rc) = 1. The corresponding result for n = 1 is an unpublished result of R. Sharma. If ,c = 0, Men+i is locally En+1 x Sn(4), as we have seen, and is tangent to the Euclidean factor giving dimN(0) = n+ 1. If rc = 1, the structure is Sasakian. This leaves the question of the dimension of N(1) on a Sasakian manifold; F. Gouli-Andreou and the

author conjecture that it must be either 2n + 1 (and Men+' is of constant curvature), or 3 (and Men+i has a Sasakian 3-structure, see Chapter 13) or 1. We just mentioned ?l-Einstein contact metric manifolds and a few remarks about this and the Einstein case are in order. First note that on a SasakianEinstein manifold Q = 2nd and Q = 2+1+1.I and hence the scalar curvature is 7- = 2n(2n + 1) > 0. An important recent result is the following Theorem of Boyer and Galicki [2001]. This remarkable result should be compared with the Goldberg conjecture that a compact almost Kahler-Einstein space is Kahler (see Section 10.2). The Goldberg conjecture is true for non-negative scalar curvature (Sekigawa [1987]). Since Sasakian-Einstein manifolds have positive

scalar curvature, one might hope for a Goldberg conjecture type result in contact geometry and Boyer and Galicki achieved the following.

Theorem 7.10 A compact K-contact Einstein manifold is Sasakian. There are many Sasakian-Einstein manifolds as shown by Boyer and Galicki [2000] including the well known Sasakian-Einstein structure on S3 X S2 (see Tanno [1978b] and Section 9.2). This paper of Boyer and Galicki contains a number of nice results on Sasakian-Einstein manifolds. It is also known that a contact metric Einstein manifold of dimension > 5 with belonging to the n-nullity distribution is Sasakian (Tanno [1988]).

If M is a compact q-Einstein K-contact manifold with Ricci tensor, p = ag + brl ®rl, and if a > -2, then M is Sasakian (S. Morimoto [1992], Boyer and Galicki [2001]).

Turning to other curvature conditions we first prove the following result of Tanno [1967b] on K-contact manifolds; the corresponding result in the Sasakian case was given by Okumura [1962a].

Theorem 7.11 A locally symmetric K-contact manifold is of constant curvature +1 and Sasakian. Proof. Since the structure is K-contact, Rgxe = -X+ii(X)e (cf. Proposition 7.1 with h = 0). Differentiation yields

R-myx +

-g(X, OY) - ?1(X )OY.

Replacing Y by ¢lY we have

RYx -

rl (Y)Rfx = g(X, Y) - 2r7(Y)r1(X )e + rl(X)Y

7.2 Curvature of contact metric manifolds

107

and simplifying

RYx + RRxY = g(X, Y) - 277(Y)X + 77(X)Y. Differentiating this gives

-RyxcZ - ROzxY = -g(X, Y)gZ - 29(Z, qY)X + g(Z, OX)Y Replacing Z by OZ and simplifying

RyxZ + RzxY = g(X, Y)Z - 2g(Z, Y)X + g(Z, X)Y. Thus finally 2g(RyxX,Y) = 21XJ2Y12 -2g(X,Y)2 giving constant curvature +1 and that the structure is Sasakian (cf. Theorem 7.3).

In [1989] the author showed that the standard contact metric structure of the tangent sphere bundle (or unit tangent bundle but with a homothetic change of metric) T1M of a Riemannian manifold M is locally symmetric if and only if either M is flat in which case T1M is locally isometric to En+1 X Sn(4)

or M is 2-dimensional and of constant curvature +1 in which case T1M is locally isometric to T1S2 N IRP3 - SO(3). We will prove this result in Section 9.2. We remark that even though T1S3 is topologically S3 X S2, the product metric is not an associated metric to the natural contact structure (again see Chapter 9 or Tanno [1978b] for an Einstein associated metric). The above results raise the question of the classification of all locally symmetric contact metric manifolds and one might conjecture that the only two possibilities are contact metric manifolds which are locally isometric to S2n+1 (1) or En+1 x Sn(4). In dimension 3, Blair and Sharma [1990b] showed that a lo-

cally symmetric contact metric manifold is of constant curvature 0 or 1. In dimension 5, A. M. Pastore [1998] showed that indeed local isometry with S5 (1) or E3 X S2 (4) are the only two possibilities. More generally she showed

that if Men+1 is a locally symmetric contact metric manifold, then Men+1 admits a foliation whose leaves are totally geodesic and locally isometric to the Riemannian product Em+1 x Sm(4), where m is the multiplicity of the +1 eigenvalue of the operator h. In [1994] K. Bang showed that a locally symmetric contact metric manifold with Rx C = 0 is locally isometric to En+1 X Sn(4); this answered positively a question posed by D. Perrone [1992a, p. 148]. Also in dimension > 3 Sharma and Koufogiorgos [1991] showed that a locally sym-

metric contact metric manifold such that the sectional curvatures of plane sections containing are equal to a constant c is either of constant curvature +1 and Sasakian or c = 0. In the same paper they showed that if belongs to the k-nullity distribution and the contact metric structure has parallel Ricci tensor, then the contact metric manifold is either Sasakian and Einstein or locally isometric to E1+1 X Sn(4).

7. Curvature of Contact Metric Manifolds

108

Similarly one can ask the question: When is a contact metric structure conformally flat? In [1962a] Okumura showed that a conformally flat Sasakian

manifold of dimension > 5 is of constant curvature +1 and in [1963,1967b] Tanno extended this result to the K-contact case and for dimensions > 3.

Theorem 7.12 A conformally flat K-contact manifold is of constant curvature +1 and Sasakian. In a similar vein, Ghosh, Koufogiorgos and Sharma [2001] have shown that a conformally flat contact metric manifold of dimension > 5 with a strongly pseudo-convex CR-structure is of constant curvature +1. In dimension > 5, as we have seen, a contact metric structure of constant curvature must be of constant curvature +1 and in dimension 3 a contact metric structure of constant curvature must be of constant curvature 0 or 1. For simplicity set Rx = IX, then l is a symmetric operator. K. Bang [1994] showed that in dimension > 5 there are no conformally flat contact

metric structures with l = 0, even though there are many contact metric manifolds satisfying l = 0, see Theorem 9.14. Bang's result was extended to dimension 3 and in fact generalized by Gouli-Andreou and Xenos [1999] who showed that in dimension 3 the only conformally flat contact metric structures satisfying 0£l = 0 (equivalent to Vch = 0, Perrone [1992a]) are those of constant curvature 0 or 1. More recently Calvaruso, Perrone and Vanhecke [1999] showed that in dimension 3 the only conformally flat contact metric structures for which is an eigenvector of the Ricci tensor are those of constant curvature 0 or 1. In the case of the standard contact metric structure on the tangent sphere bundle, the metric is conformally flat if and only if the base manifold is a surface of constant Gaussian curvature 0 or 1 (Theorem 9.7) as was shown by Blair and Koufogiorgos [1994]. In view of these strong curvature results one may ask if there are any conformally flat contact metric structures which are not of constant curvature. We devote the rest of this section to showing the local existence and giving some additional remarks. We will work in R3, primarily with cylindrical coordinates (r, 0, z). Let rl = a (adr +,3rdO + ydz) be a contact form on R3. Then d77 =

1

2((3+r/dr-ae)drAd8+(yr -a.,)drAdz+(ye -r,3Z)d9Adz).

If g is a conformally flat metric, we may write it as ds2 = 4e2°(dr2 + r2d02 + dz2). If g is also an associated metric, the characteristic vector field is given by

2e-2, and

1 gives e2Q = a2 + ,32 ,+ ,y2.

0r1

+yBzJ

7.2 Curvature of contact metric manifolds

109

Much of our analysis will actually be done with respect to the Euclidean metric on ]Ig3 and we denote the Euclidean length of a vector field B by IBI. In particular if B = as +i3-. ae +''ai, IBI = e2° . Now dri(e, X) = 0 for all X gives 1

1/(a+rO,-ae)+ry(YT-a=)0, -r,C3=)=0, 1

a(az -'Yr) + /3(r/3z -'ye) = 0

and therefore curl B is proportional to B, say curl B = f B. Computing 0, using 02 = -I + r, ® and the fact that 77 A dr7 0, one finds that f2 = e2a. Therefore curl B = ±e°'B. Since we may change the sign of 77 without changing

the problem, our study reduces to the study of the differential equation

cur1B = IBIB. Thus we are at the point that the existence of 3-dimensional conformally flat contact metric manifolds corresponds to finding solutions of cur1B = IBIB. Had we carried out our analysis in cartesian coordinates (x, y, z) we would have been led to the same differential equation for a vector field B = The flat case corresponds to the familiar unit vector field B = sin z -L +cos z ay

that is equal to its own curl. For the standard Sasakian structure of constant curvature +1 on S3, using stereographic projection to R3, the corresponding vector field satisfying curlB = IBIB is 4(1 + z2 - x2 - y2) 8

8(xz - y) 8 8 8(x + yz) B= (1+x2+y2+z2)28x+(1+x2+y2+z2)28y+(1+x2+y2+z2)28z

In neither of the two constant curvature examples above are the component

functions, functions of the radial coordinate alone. If cur1B = IBIB and the component functions are functions of r alone, then a = 0 and the other components, O(r) and -y(r), satisfy

02+,y2,y

-,y

02+'Y20.

(*)

Local existence away from r = 0 follows easily from the standard existence theorems for systems of ordinary differential equations. Finally one can show that the metric ds2 = 4e2° dr2+r2d02+dz2) with e2, _ /32 + y2 is not of constant curvature. Differentiating e2° = X32 + ry2 using (*) we 2

have Q = - 132v . If g were of constant curvature the value of the curvature

would be 0 or +1. Consider the unit vector field X =

2e-2°

(r--

a

TO -

A

1

8z1

110

7. Curvature of Contact Metric Manifolds

belonging to the contact subbundle D. Then computing the sectional curvature of the plane section spanned by and X we have g(Rxgg, X) =

-4e-2a (./2 +

Q) =

r

-6a

4e -6a

r2

o2y2.

If the constancy of the curvature were 0, then ,G or y would vanish and from (*) both would vanish making 77 .= 0, a contradiction. If the curvature is +1, re'" = 2/3y; differentiating this using (*) and a' _ p2o we again have that

J3=0. The equation curlB = aB for some function a arises in solar physics and cur1B = JBIB can be viewed as a special case. An active region on the sun is an area of extreme magnetic flux. These regions contain sunspots as well as the highly volatile phenomena of solar flares. For some solar phenomena, such as flares and prominences, the Lorentz force, j x B, j being the current density, dominates the pressure gradient and gravitational forces and thus for a relatively slow moving plasma we have from the equations of motion the approximation j x B = 0, the so-called "force-free" field. From Maxwell's equa-

tions we have, for an electrically neutral field, curl B = aj. Thus one is lead to curlB = aB. There are many solutions in the literature (see e.g., Priest [1982]) which by Maxwell's equation must also satisfy divB = 0. Except for the simple case B = sinze5 + coszey, the solutions listed in the literature do not satisfy curlB = BIB and we mention only a couple of them. The Lundquist solution (see e.g., Salingaros [1990] or Priest [1982, p. 138] for a gen-

eralization) is given in cylindrical coordinates by B = (0, J1(r), Jo(r)), Jo and Jl being Bessel functions, satisfies curl B = B but it is not of unit length. Another common solution is B = --- (ae + az) which satisfies curlB = 21B12B (Priest [1982, p. 262] where there are of course additional physical constants). While our vector field is another solution and one whose series expansion converges rapidly, so that its calculation for physical purposes is not prohibitive, the matter of whether the equation for a "force-free" model is even physically appropriate is questionable (see e.g., Salingaros [1986,1990]).

7.3

q5-sectional curvature

In this section we introduce the notion of 0-sectional curvature, this idea plays the role in Sasakian geometry that holomorphic sectional curvature plays in Kahler geometry. A plane section in TmM2n+1 is called a c,-section if there exists a vector X E T,,,,M2n+i orthogonal to such that {X, OX} span the section. The sectional curvature K(X, OX), denoted H(X), is called 0-sectional curvature.

7.3 0-sectional curvature

111

Recall that the sectional curvatures of a Riemannian manifold determine the curvature transformation RxyZ. It is also well known that the holomorphic sectional curvatures of a Kahler manifold determine the curvature completely. We shall show that on a Sasakian manifold the 0-sectional curvatures determine the curvature completely (Moskal [1966] ). Let B(X, Y) = g(Rx yY, X) and for X orthogonal to , let D(X) = B(X, OX). Also recall the tensor field P defined by P(X, Y, Z, W) = drl(X, Z)g(Y, W) - drl(X, W)g(Y, Z)

-drl(Y, Z)g(X, W) + drl(Y, W)g(X, Z) in Section 7.1.

Proposition 7.8 On a Sasakian manifold, for tangent vectors X and Y orthogonal to e we have

B(X, Y) = 32 (3D(X + OY) + 3D(X - OY) - D(X +Y) - D(X - Y) -4D(X) - 4D(Y) - 24P(X, Y, X, OY)). Proof. Direct expansion and Lemma 7.1 give 32 (3D(X +

OY) + 3D (X - OY) - D(X + Y) - D(X - Y)

-4D(X) - 4D(Y) - 24P(X,Y, X, OY))

= 32 (6g(RxyY, X) + 6g(Rox 0y0Y, OX) + 8g(Rx 0x0Y, Y) +12g(RxyoY,,OX) -2g(RxOyOY,X) - 2g(RmxyY,OX) +4g(Rx,yY, OX) - 24P(X, Y, X, OY))

=

g(RxyY,X).

Proposition 7.9 Let M be a Sasakian manifold and {X, Y} an orthonormal pair in TM with X and Y orthogonal to . Set g(X, OY) = cos 0, 0 < 0 < 7r. Then the sectional curvature K(X, Y) is given by K(X, Y) = 8 (3(1 + cos 0)2H(X + OY) + 3(1- cos 0)2H(X - OY)

-H(X + Y) - H(X - Y) - H(X) - H(Y) + 6 sine 0).

7.

112

Curvature of Contact Metric Manifolds

Proof. K(X, Y) = B(X, Y) in the previous proposition, so we examine the terms in the expansion of B(X,Y). Clearly D(X) = g(X,X)2H(X) for any X orthogonal to , and hence for the given pair {X, Y}, g(X +OY, X +OY) = 2(1 +cos B), g(X - OY, X - OY) = 2(1 - cos B), g(X + Y, X + Y) = 2 and g(X - Y, X - Y) = 2. Thus D(X + OY) = 4(1 + cos 9)2H(X + OY), and so on. Finally note that P(X, Y, X, OY) = - sine 6 completing the proof. Theorem 7.13 The 0-sectional curvatures of a Sasakian manifold determine the curvature completely.

Proof. Since the sectional curvatures of a Riemannian manifold determine the curvature, it suffices to show that for an orthonormal pair {X, Y}, K(X, Y)

is determined uniquely by H and g. If X and Y are orthogonal to , the previous proposition applies. If X or Y is 6, K(X, Y) = 1. So suppose that X = 77(X)6+aZ and Y = 77(Y)6+bW where 77(X), 77(Y), a = N/77(X)2 and b = 1 - r7(Y)2 are non-zero. Recall that on a Sasakian manifold Rgz6 = -Z and RzEZ = - for any unit vector orthogonal to . Therefore bW, 77(X) + aZ)

K(X, Y) = =

b277(X)2 - 2abr7(X)r7(Y)g(Z, W) + a277(Y)2 + a2 b2g(Rz wW, Z).

Now g(Z, W) + abg(X - 77(X )e, Y -

9(RzwW,Z)= (1-9(Z,W)2)K(Z,W)

b77(X)71(Y) and so

a1 77(X)271(Y)2)K(Z,W).

Thus

K(X, Y) = 7)(X)2(1 - 77(Y)2) + 2,q(X)277(Y)2 + 77(')2(1

- 77(X)2)

+((1 _,q(X)2)(1 - 77(Y)2) - 77(X)271(Y)2))K(Z, W) = 77(X)2 +

77(Y)2

+ (1- 77(X)2 - 77(Y)2)K(Z, W)

and K(Z, W) is given by the previous proposition completing the proof.

Note that the above proof uses not only the values of the 0-sectional cur-

vatures, but also the facts that on a Sasakian manifold RgX = -X and for any unit vector X orthogonal to 6. Thus we have actually proved that any tensor field of type (1,3) on a Sasakian manifold which satisfies the symmetries of the curvature tensor, the first Bianchi identity, identity

a) of Lemma 7.1, RgX = -X and RX gX = - for any unit vector X orthogonal to and which agrees with the values of the 0-sectional curvatures must be the curvature tensor. Therefore we can easily prove the following theorem of Ogiue [1964].

7.3 0-sectional curvature

113

Theorem 7.14 If the O-sectional curvature at any point of a Sasakian manifold of dimension > 5 is independent of the choice of q5-section at the point, then it is constant on the manifold and the curvature tensor is given by

RxyZ = c

3

MY, Z)X - g(X, Z)Y)

4 +c 4 I (77(X)q(Z)Y - 77(Y)?7(Z)X + g(X, Z)rl(Y) - g(Y, Z)71(X ) +(Z, Y)OX - (D (Z, X)g5Y + 2-1) (X,Y)OZ)

where c is the constant O-sectional curvature.

Proof. In view of the above remark, in order to see that the curvature tensor has the above form with c a function on the manifold, one need only check the necessary conditions and this is easily done. The Ricci tensor p and the scalar curvature r are given by

p(X,Y) =

n(c + 3)+ c - 1g(X Y) - (n + 1)( c -1) 2

rl(X)77(Y)

and

r= 2(n(2n+1)(c+3)+n(c- 1)). Now from the second Bianchi identity, V, ,,r - 2Vsp' . = 0 where p°c, are the components of the Ricci tensor of type (1,1) and hence

(n - 1)dc +

0.

Applying this to , we have c = 0 and hence do = 0 for n

1 as desired.

A Sasakian manifold of constant O-sectional curvature c will be called a Sasakian space form and denoted M(c). Note that a Sasakian space form has constant scalar curvature and is r7-Einstein. Also if c < 1 we have the following pinching of the sectional curvature, similar to that in the Kahler case,

c 5 for

which the O-sectional curvature at any point is independent of the choice of q5-section at the point. He proved that the cl-sectional curvature is constant and obtained the curvature tensor explicitly.

114

7.4

7.

Curvature of Contact Metric Manifolds

Examples of Sasakian space forms

To begin, let (0, , r7, g) be a contact metric structure and recall the notion of a D-homothetic deformation: 1

=1

7=arl,

_0, 9=ag+a(a-1)r7®r7

i , g) is again a where a is a positive constant. The deformed structure contact metric structure and it enjoys many of the properties of the original

structure as we remarked in Section 7.2. In particular if (0, , 77, g) is Sasakian, so is (0, , i7, g); if M(c) is a Sasakian space form, then deforming the structure we obtain the Sasakian space form M(c) where c = `+-a3 - 3 (see Tanno [1968], [1969] for details). We will show that there exist Sasakian space forms M(c) for every value of c. Moreover we state the following theorem of Tanno and refer to [1969] for the proof.

Theorem 7.15 Let M(c) be a complete, simply connected Sasakian manifold with constant q5-sectional curvature c. Then M(c) belongs one of the three families of examples listed below. 7.1.1

Stn+1

Let (0, l;, 77, g) be the standard contact metric structure on the sphere S1n+1 constructed either as a hypersurface of C2n+2 (Example 4.5.2 and Section 6.3) or as a principal circle bundle over CPn (Examples 4.5.4 and 6.7.2). Applying a D-homothetic deformation to this structure one obtains a Sasakian structure on S2n+1 with constant 0-sectional curvature c = 1 - 3. Note that from the remark on pinching in Section 7.3, these metrics on S2n+1 for c > 0 satisfy the condition of Goldberg's theorem referred to in Section 6.8. 7.x.2

R2n+1

In Examples 4.5.1 and 6.7.1 we saw that R2n+1 with coordinates (xi, yi, z), i = 1, ... , n admits the Sasakian structure n

(dz -

y= 2

n

yidx2), i=1

g = 77 ®77 +

E((dxi)2 + (dyZ)2). 4 i=1

With this metric, R2n+1 is a Sasakian space form with c = -3 (cf. Okumura [1962b]).

7.5 Locally 0-symmetric spaces

115

74.3 Bn x R Let Bn be a simply connected bounded domain in Cn and (J, G) a Kahler structure with constant holomorphic sectional curvature k < 0. Since the fundamental 2-form S2 of the Kahler structure is closed, Sl = dw for some real analytic 1-form w. Now on Bn x R let it denote the projection onto Bn and t the coordinate on R. Then Bn x R with 77 _ 7r*w + dt and g = ,7r*G-I-77®?7 is a Sasakian manifold. Regarding g as a connection form on Bn xR,

let FrX denote the horizontal lift of a vector field X on Bn. Also we denote by K the sectional curvature of Bn. Then by direct computation (see Ogiue [1965] or use the Riemannian submersion technique of O'Neill [1966]), we have

K(irX, irY) = K(X, Y) - 317(VsxirY)2 where {X, Y} is an orthonormal pair -g(FrY JX). on Bn. Now Therefore Bn x R has constant 0-sectional curvature c = k - 3.

7.5

Locally 0-symmetric spaces

We saw in Theorem 7.11 that a locally symmetric K-contact manifold is of constant curvature +1 and Sasakian and we raised the question of the classification of locally symmetric contact metric manifolds. Certainly for Sasakian or K-contact geometry, Theorem 7.11 can be regarded as saying that the idea of being locally symmetric is too strong. For this reason T. Takahashi [1977] introduced the notion of a locally 0-symmetric space. A Sasakian manifold is said to be a Sasakian locally 0-symmetric space if q12(VvR)xyZ = 0

for all vector fields X, Y, Z orthogonal to ; such spaces were called locally D-symmetric spaces by Shibuya [1982]. It is easy to check that Sasakian space forms are locally 0-symmetric spaces. In [1987a] Blair and Vanhecke showed that a Sasakian manifold is locally 0-symmetric if and only if

g((OxR)xOxX, OX) = 0 for all vector fields X orthogonal to . Note that on a Sasakian manifold M, or more generally on a K-contact manifold, a geodesic that is initially orthogonal to 1; remains orthogonal to l;. We call such a geodesic a 0-geodesic. A local diffeomorphism sm, of M, m E M,

is a 0-geodesic symmetry if its domain contains a (possibly) smaller domain U such that for every 0-geodesic y(s) parametrized by arc length we have i) y(0) is in the intersection of U and the integral curve of through m, and ii) (Sm 0 -Y) (S) ='Y(-S)

116

7.

Curvature of Contact Metric Manifolds

for all s with 7(+s) E U. Since the points of the integral curve of we have m are fixed, we see that setting S = -I + 277

s, =

through

o Sm o exp.ml.

Takahashi [1977] defines a Sasakian manifold to be a Sasakian globally ¢symmetric space by requiring that any 0-geodesic symmetry can be extended to a global automorphism of the structure and that the Killing vector field generates a 1-parameter group of global transformations. I

Among the main results of Takahashi [1977] are the following four theorems,

Theorem 7.16 A Sasakian locally 0-symmetric space is locally isometric to a Sasakian globally 0-symmetric space and a complete, connected, simplyconnected Sasakian locally 0-symmetric space is globally 0-symmetric.

Theorem 7.17 A Sasakian manifold is locally 0-symmetric if and only if it admits a q-geodesic symmetry at every point which is a local automorphism of the structure. Now suppose that U is a neighborhood on M on which is regular; then since M is Sasakian, the projection 7r : U -* V = U/ gives a Kahler structure on V. Furthermore, if denotes the geodesic symmetry on V at 7r(m), then s,(,,n) 0 7r = 71 0 Sm.

Theorem 7.18 A Sasakian manifold is locally 0-symmetric if and only if each Kahler manifold which is the base of a local fibering is a Hermitian locally symmetric space.

Following Okumura [1962a] define on a Sasakian manifold a linear connec-

tion V by V Y = V xY + TxY where TXY = drl(X, Y) - 77(X)gY + rl(Y)OX

and it is easy to see that the structure tensors are parallel with respect to th connection. Takahashi [1977] then proves the following.

7.5 Locally

spaces

117

Theorem 7.19 A Sasakian manifold is a locally 0-symmetric space if and only if VR = 0, equivalently

(VvR)xyZ = -TvRxyZ + RTVxYZ + RxTVYZ + RxyTvZ for all X, Y, Z, V.

In the spirit of the fact that a Riemannian manifold is locally symmetric if and only if the local geodesic symmetries are isometries and in view of the above results of Takahashi, we state the following extension of Theorem 7.16 (sufficiency in the Sasakian case, Blair and Vanhecke [1987b]; in the K-contact case, Bueken and Vanhecke [1989]).

Theorem 7.20 On a Sasakian locally 0-symmetric space, local 0-geodesic symmetries are isometries. Conversely if on a K-contact manifold the local 0-geodesic symmetries are isometries, the manifold is a Sasakian locally symmetric space.

The notion of a locally 0-symmetric space has been for the most part explored only in the Sasakian context and it is not clear what the corresponding notion should be for a general contact metric manifold. Without the K-contact property one loses the fact that a geodesic, initially orthogonal to remains orthogonal to 1. We have just seen that in the Sasakian case local (l-symmetry is equivalent to reflections in the integral curves of the characteristic vector field being isometries. Boeckx and Vanhecke [1997] proposed this property as the definition for local 0-symmetry in the contact metric case and formalized two notions in Boeckx, Bueken and Vanhecke [1999]. We adopt this formulation here. A contact metric manifold is a weakly locally 0-symmetric space if it satisfies q12(VvR)xyZ = 0 for all vector fields X, Y, Z orthogonal to as in the Sasakian case. A contact metric manifold is a strongly locally 0-symmetric space if reflections in the integral curves of the characteristic vector field are isometries. From Chen and Vanhecke [1989] (see also Boeckx, Bueken and Vanhecke [1999]) one sees that on a strongly locally 0-symmetric space

9((OX...xR)xyX, ) = 0,

g((OXxR)xyX, Z) = 0, 0, 9((V . for all X, Y, Z orthogonal to l; and all k E N. Conversely on an analytic Riemannian manifold these conditions are sufficient for the contact metric manifold to be a strongly locally 0-symmetric space. In particular taking k = 0

118

7.

Curvature of Contact Metric Manifolds

in the second condition, we note that a strongly locally 0-symmetric space is weakly locally 0-symmetric. Calvaruso, Perrone and Vanhecke [1999] determine all 3-dimensional strongly locally 0-symmetric spaces. Examples of strongly locally O-symmetric spaces include non-Sasakian contact metric manifolds for which belongs to the (rc, µ)-nullity distribution (Boeckx [1999]). Special cases of these are the non-abelian 3-dimensional unimodular Lie groups with left-invariant contact metric structures (Boeckx, Bueken and Vanhecke [1999]). Boeckx, Bueken and Vanhecke [1999] also gave an example of a non-unimodular Lie group with a weakly locally 0-symmetric

contact metric structure which is not strongly locally 0-symmetric. To see these last examples explicitly we include the classification of simply connected homogeneous 3-dimensional contact metric manifolds as given by Perrone [1998]. Let W = s (r-Ric(h)+4) denote the Webster scalar curvature (cf. Section 10.4 below). The classification of 3-dimensional Lie groups and their left invariant metrics was given by Milnor in [1976].

Theorem 7.21 Let (M3, p, g) be a simply connected homogeneous contact metric manifold. Then M is a Lie group G and both g and O are left invariant. More precisely we have the following classification: (1) If G is unimodular, then it is one of the following Lie groups:

1. The Heisenberg group when W =

gj = 0;

2. SU(2) when 4VW > I k£gI ;

S. the universal covering of the group of rigid motions of the Euclidean

plane when 4'W = I'egI > 0;

4/W
1 - 2(n+2 then Men is totally geodesic.

In his excellent survey article [1974], Ogiue obtained other results, posed several conjectures and open problems, and continued this theme in [1976a].

124

8. Submanifolds of Kahler and Sasakian Manifolds

For example in the context of the above results Ogiue proved in [1976a] that if the sectional curvature K of Men exceeds n3 or if K > s and H > 2, then M2n is totally geodesic. Ogiue's conjectures from [1974] and [1976a] included that i) H > 2 or ii) K > s and n >_ 2 imply that Men is totally geodesic. In [1985a] A. Ros introduced a new technique to attack this kind of problem for Men compact, viz., for unit tangent vectors V regard I a(V, V) I2 as a function of the unit tangent bundle of M2n and to study its behavior at its maximum point. The result of Ros in [1985a] is that Ogiue's conjecture i) is true and using the same technique Ros and Verstraelen in [1984] proved conjecture ii). 3 Notice that the Calabi (Veronese) imbeddings of CPn(2) into CPn 2 (1) show that these conjectures are best possible (see Section 12.7). In fact in [1985b] Ros gave a complete classification of compact Kahler submanifolds of CPn+p(1) with H > 2; there are seven interesting cases. As an illustration of the Ros technique we will give in the next section the proof of a theorem of Urbano [1985] on Lagrangian submanifolds of OP'. The corresponding problem in Sasakian geometry is to study a compact invariant submanifold Men+1 of a Sasakian space form M2(n+p)+1(c) with constant c-sectional curvature c > -3. This was first taken up by Harada [1973a], [1973b], Kon [1976] and later by VanLindt, Verheyen and Verstraelen [1986], Again the later results are the stronger ones and use the Ros technique; in particular VanLindt, Verheyen and Verstraelen prove that if the q5-sectional curvature of Men+1 exceeds `23 or if the sectional curvature of Men+1 exceeds c 83 then Men+i is totally geodesic.

8.2

Lagrangian and integral submanifolds

We have already seen in Section 1.2 that the maximum dimension of an isotropic submanifold of a symplectic manifold Men is n and that an ndimensional isotropic submanifold is called a Lagrangian submanifold. In Section 5.1 we saw that the maximum dimension of an integral submanifold of a contact manifold Men+1 is also n. In almost Hermitian geometry, isotropic submanifolds are known as totally real submanifolds since they are characterized by the fact that the almost complex structure maps the tangent space at any point into the normal space at the point.

In the spirit of the results of the last section we mention a few results on compact minimal Lagrangian submanifolds Mn of a complex space form M2n(c). In [1974] Yau considered a totally real minimal surface M2 in a Kahler surface of constant holomorphic curvature c and proved the following results: 1) If M2 has genus zero, M2 is the standard imbedding of 1RP2 in (CP2. 2) If

M2 is complete and non-negatively curved, it is totally geodesic or flat. 3) If M2 is complete and non-positively curved with Gaussian curvature K and if

8.2 Lagrangian and integral submanifolds

125

4 -K > a > 0 for some constant a, then M2 is totally geodesic or flat. In [1973] Houh proved that if a totally real minimal surface M2 in CP2 has constant scalar normal curvature, it is totally geodesic or is non-positively curved and, combining with Yau's result, Houh showed that if in addition M2 is complete, it is totally geodesic or flat. The flat case is realized by T2 imbedded in CP2 as a flat minimal Lagrangian submanifold, Ludden, Okumura and Yano [1975].

In [1976b] Ogiue showed that if c > 0 and the sectional curvature K of M" satisfies K > a(2-2), , then Mn is totally geodesic. This was extended n-2), by Chen and Houh [1979] who showed that if c > 0 and K > 4(2n-1) , then either Mn is totally geodesic or n = 2 and the surface M2 is flat. If a minimal Lagrangian submanifold in M2, (C) is itself of constant curvature k, then it was shown by Chen and Ogiue [1974a] that the submanifold is either totally geodesic or k < 0; this was improved upon by Ejiri [1982] who showed that the submanifold is either totally geodesic or flat. Finally to give a stronger result and to illustrate the technique of A. Ros, we give the following theorem of Urbano [1985] with proof.

Theorem 8.4 Let M be a compact Lagrangian submanifold minimally immersed in CPn(c). If the sectional curvature K of M is greater than 0, then M is totally geodesic.

Proof. We denote by (J, g) the almost Hermitian structure on CPn(c) where g is the Fubini-Study metric of constant holomorphic curvature c. For tangent vectors X and Y we have readily

0 = (7xJ)Y = VxJY - JOxY = -AJYX + VXJY - JV Y - JQ(X, Y) giving

AJYX = -Ju(X, Y),

V JY = JVxY.

The equations of Ricci-Kiihn and Gauss give

R'(X,Y, JZ, JW) = g(RxyJZ, JW) +g([Ajz,AJw]X,Y) = g(RxyZ,W)) - 9(a(Y, Z), a(X,W)) +9(a(X, Z), a(Y,W)) +g(AjwX, AjzY) - g(AJZX, AjwY)

= g(RxyZ,W) Since the ambient space is of constant holomorphic curvature, RxyZ = 4(g(Y,Z)X-g(X,Z)Y-Q(Y,Z)JX+Q(X,Z)JY+2Sl(X,Y)JZ) and hence

that (RxyZ)1 = 0. Thus by the Codazzi equation (V'a)(X,Y,Z) V a(Y, Z) - a(VxY, Z) - a(Y, VxZ) is symmetric. Defining V'2a by (V,2a)

(X, Y, Z, W) = V ((V u) (Y, Z, W) )

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8. Submanifolds of Kahler and Sasakian Manifolds

-(V'o)(VxY, Z,W) - (V'o)(YVxZ,W) - (V'o)(Y Z,VxW) we have by the symmetry of (V'o)(X, Y, Z) that (V,2o)(X,Y,

Z, W) = (V12a)(Y,X, Z, W) +RxYo(Z,W)

-cr(RxyZ, W) - v(Z, RxyW). Define a real-valued function on the unit tangent bundle T1M by f (V) _ g(o(V, V), JV). Since T1M is compact, f attains its maximum at a unit vector V tangent to M at a point p. For any unit tangent vector U at p, let 'T(t) be

the geodesic in M with -y(O) = p and 7'(0) = U. Let V (t) be the parallel vector field along 'y with V(O) = V. Then 0

dtf(V(t)) t=o = g((V'o)(U,V,V), JV).

Also

0> dt2f(V(t))I t=o =

g((V'2a)(U,U,VV),

= 9((Vr2a)(V, U, V, U) + RU -L

JV) = g((Vi2g)(U,VV, U), JV)

U) - v(RuvV, U) - o,(RuvU, V), JV)

= ((Vi20')(V,V, U, U), JV) +g(RuvV, Jo(U,V))

-g(AjuRuvV, V) - g(AivRu vU, V) = ((V'2Q)(1' V, U, U), JV) + 2g(RuvV, Jo- (U, V)) + g(RuvU, Jo- (V, V)).

On the other hand restricting f to the fibre of T1M at p and taking U orthogonal to V we have

0 = U f = 2g(a(U, V), JV) + g(o (V, V), JU) = 2g(AjvV, U) + g(u(V,V), JU) = 3g(o-(V, V), JU). Therefore since U is any unit vector orthogonal to V, cr(V, V) = f (V)JV or AjVV = f (V)V,

i.e., V is an eigenvector of AJV with eigenvalue f (V). Furthermore

0> U2f =6g(o-(U, V),JU)+3g(u(V,V),J(-V)) = 6g(u(U, U), JV) - 3f (V) = 6g(AivU, U) - 3f (V).

8.2 Lagrangian and integral submanifolds

127

Thus if {U1, ... , U} is an orthonormal eigenvector basis of Ajv with U. = V and if a2, i = 1, ... , n - 1 the other eigenvalues,

f(V)-2,2>0. Now from the above and the minimality n

o > E {9((O2U)(V, V,

Uz,

Ui), JV)

z-1

+2g(Rui vV, Jo, (U,, V)) +g(RuuvUj, Ja(V,V))} n-1

-2AiK(V,Uz) + f(V)K(V,UU)} i=1

K(V,UZ){f(V)-2);,}. =1

Finally since the sectional curvature is positive, we conclude that each X;, =

'f (V) and hence trAjv =

(n+12fv

= 0 giving f (V) = 0. Now f (-U) =

- f (U) and V was the maximum point of f, so f = 0. Thus o(V, V) is orthogonal to JV and to JU for any U L V and hence v(V, V) = 0 for any vector V and so M must be totally geodesic. In Section 1.2 and Example 5.3.3 we remarked that there is no topological imbedding of a sphere as a Lagrangian submanifold of Cn and no umbilical, non-totally-geodesic, Lagrangian submanifolds of a complex space-form. The immersed Whitney sphere was the closest candidate with only one double point. Its second fundamental form v was given by n+2(g(X,Y)H+g(JX,H)JY+g(JY,H)JX)

Q(X,Y)

where H denotes the mean curvature vector (Borrelli, Chen and Morvan [1995], Ros and Urbano [1998]). More generally a Lagrangian submanifold of a Kahler manifold is said to be Lagrangian H-umbilical if the second fundamental form a is of the form

a(X, Y) = ag(JX, H)g(JY, H)H +/3g(H, H)(g(X,Y)H + g(JX, H)JY + g(JY, H)JX) for suitable functions a and 0. This notion was introduced and a classification of such submanifolds in complex space forms was given by Chen in a series of papers [1997a,b],[1998].

128

8. Submanifolds of Kahler and Sasakian Manifolds

In the contact case, integral submanifolds are often called C-totally real submanifolds since 0 maps the tangent space at any point into the normal space. However, we will not adopt this term here. As a reminder, since ?7(X) _ 0 for any vector X tangent to the integral submanifold, is a normal vector field.

Examples that we might mention here are Sn C S2n+1 as a totally geodesic integral submanifold as was described in Example 5.3.1 and T2 C S5 as a flat minimal integral submanifold as was described in Example 5.3.2. We begin with the following lemma.

Lemma 8.1 Let M be an integral submanifold of a K-contact manifold M. Then A£ = 0.

Proof. g(A4X, Y) = 9(o(X, Y), ) = 9(0x1', ) = -9(Y,

-9(Y, OX) = 0.

e Now for an integral submanifold Mn of a Sasakian manifold Men+1, let {el, ... en} be a local orthonormal basis on M. Then {oe1, ... , qlen, } is an ,

orthonormal basis of the normal space at each point of the local domain. For simplicity we write Ai for A0e; .

Lemma 8.2 Let M be an integral submanifold of a Sasakian manifold M. Then Aie; = A;ei.

Proof. g(Aiej, ek) = g(a(ei, ek), Oei) = -g(e7,

(De,,/ )ei

+ 0Dekei) = g(De, ei, Oe.i) = g(Ajei, ek).

Now let Mn be an integral submanifold of a Sasakian space form M(c); the Gauss equation yields

g(RxyZ, W) = c

3

(g(X, W)g(Y, Z) - g(X, Z)g(Y, W))

4 (g(AiX, W)g(AiY, Z) - g(AiX, Z)g(AiY, W)) i=1

by virtue of Lemma 8.1. In turn the sectional curvature K(X, Y) of Mn determined by an orthonormal pair {X, Y} is given by 3

K(X, Y) = c

4

+ (g(AiX, X)g(AiY, Y) - g(AiX, Y)2). i=1

8.2 Lagrangian and integral submanifolds

129

Moreover the Ricci tensor p and the scalar curvature T of Mn are given by

P(X,Y) =

n-1 4

" " (c + 3)g(X, Y) + > '(trAi)g(AiX, Y) - > ' g(AiX, AiY), i=1

i=1

" T = n(n 4

1)

(c + 3) + 7 (trAi)2) - O 2. i=1

From these expressions the following proposition is not difficult and we omit the proof.

Proposition 8.1 Let M" be an integral submanifold of a Sasakian space form M2"+1 (c) which is minimally immersed. Then the following are equivalent: a) Mn is totally geodesic.

b) Mn is of constant curvature c43

c) p = 4(c + 3)g. d) T = " 4 1 (c + 3).

Similar to the result of Chen and Ogiue [1974a] in the Kahler case, Yamaguchi, Kon and Ikawa [1976] proved the following result. Theorem 8.5 Let Mn be a minimal integral submanifold of a Sasakian space form M2"+1 (c). If Mn has constant curvature k, then either M" is totally geodesic or k < 0. For the standard Sasakian structure on S5(1) we give the following theorem of Yamaguchi, Kon and Miyahara [1976] and we include a proof since its techniques, though not new, are different than those presented so far in this book.

Theorem 8.6 Let M2 be a complete integral surface of S5(1) which is minimally immersed. If the Gaussian curvature K of M2 is < 0, then M2 is flat. Proof. Choose a system of isothermal coordinates (x1, x2) so that the induced metric g is given by g = E((dxi)2 + (dx2)2). Let Xi = a;; and aij = o,(Xi, X;). Recall the standard formulas for the induced connection:

Vx,X1 = -px2X2 =

XjE 2E

X1

2E X2,

2E X1 + 2E X2.

Using the minimality and the Codazzi equation, one readily obtains vXi0-12

- VX2o'11 = 0,

VXiall + V42Q12 = 0.

8. Submanifolds of Kahler and Sasakian Manifolds

130

Now define a complex-valued function F by F = 9(911, OXi) - 29(912, OX1)

Note that F is nowhere zero on M2, for if F = 0 at some point m, then g(au, OX1) and g(a12i OXl) vanish at m, but by minimality g(0-22, OX1) = -g(a11i qX1) = 0 and by Lemma 8.2 g(a12, OX2) = 9(0'22, OX1) = 0 and -§(922, 95X2) = §(911, glX2) = g(0'12i O X1) = 0. Thus a vanishes at m and so by the Gauss equation, the Gaussian curvature at m is +1 contradicting the hypothesis K < 0. Differentiating the real part of F with respect to X1 we have r X12EEOX1 X1RF = g(OXi911, 0X1) + 9'(911, ES + X2EEOX1

_ -g(V 2a12, OX1) + g(1711,

-

-

X222,E

X22E,E

OX2)

6X2)

Differentiating with respect to X2 and making similar calculations for the imaginary part of F, Lemma 8.2 and the minimality yield X1IQF = X2 F and X2SlJ F = -X1sF. Thus F is analytic and therefore log IFI2 is harmonic. Now IF12 = g(a11, cX1)2+g(0-12, OX1)2. On the other hand the Gauss equa-

tion gives the Gaussian curvature K as K = 1 + E2 (g (911, 922)) - (0'12, 912) = 1 - E3 IFI2.

Thus IF12 = E3 (12x) Note also the classical formula for the Gaussian curvature of g, namely K = 6E A log E3. Suppose now that the Gaussian curvature of M2 is non-positive. Then OlogIEF2

I

=-A log E3=6EK -3. If K > 0, then Mn is totally geodesic.

If the sectional curvature is only > 0, one can do better in dimension 7, namely, we have the following result of Dillen and Vrancken [1989].

Theorem 8.8 Let M3 be a compact integral submanifold of the standard Sasakian structure on S'(1) which is minimally immersed. If K > 0, then either M3 is totally geodesic, M3 is a covering of the 3-torus or M3 is a covering of S' (/) x S2 (2) . In dimension 5 the third case does not have an analogue and the corresponding result was given by Verstraelen and Vrancken [1988]. For the example of Sl (f) x S2(,1-3) in the above theorem, the sectional curvatures satisfy

0 < K < s where both extremal values are attained (Dillen and Vrancken [1990]). Restricting the curvature to 0 < K < 1, Dillen and Vrancken proved in [1990] the following theorem.

Theorem 8.9 If M' is a compact minimal integral submanifold of S2n+1(1) and if 0 < K < 1, then K is identically 0 or 1.

Other conditions on integral submanifolds of Sasakian space forms that have been considered include the notions of the mean curvature vector H and second fundamental form being C-parallel. That is, VXH is parallel to for all tangent vectors X and respectively, (V'a) (X, Y, Z) is parallel to for all tangent vectors X, Y, Z. Recall also that a curve y(s) parametrized by are length in a Riemannian manifold is a Frenet curve of osculating order r if there exist orthonormal vector fields El, E2, ... , Er along -y, such that 'Y = El, V E1 = k1E2, V. E2 = -k1E1 + k2E3, ... , OryEr_1 = -kr_2Er_2 + kr_iEr, V Er = -kr_1Er_1 where k1, k2, ... , kr_1 are positive C°° functions of s. k; is called the j-th curvature of y. So, for example, a geodesic is a Frenet curve of osculating order 1, a circle is a Frenet curve of osculating order 2 with k1 a constant; a helix of order r is a Frenet curve of osculating order r, such that k1, k2, ... , kr_1 are constants. With these ideas in mind Baikoussis and Blair [1992] proved the following theorem.

Theorem 8.10 Let M2 be an integral surface of a Sasakian space form M5(c).

If the mean curvature vector H is C-parallel, then either M2 is minimal or locally the Riemannian product of two curves as follows: (i) a helix of order

132

8. Submanifolds of Kahler and Sasakian Manifolds

4 and a geodesic or helix of order 3, (ii) a helix of order 3 and a geodesic or helix of order 3, or (iii) a circle and a geodesic or helix of order 3. On the other hand Baikoussis, Blair, and Koufogiorgos [1995] obtained the following result.

Theorem 8.11 Let M3 be an integral submanifold of a Sasakian space form M7(c) with C-parallel second fundamental form. Then locally M3 is either flat or totally geodesic or a product 7 x M2, where -y is a curve and M2 is a surface of constant curvature and also has C-parallel second fundamental form.

We close this section with an example of this theorem in the case of 0sectional curvature c < -3. In Example 7.4.3 we saw that the product B" x R, where B" is a simply connected bounded domain in Cn, has a Sasakian structure of constant 0-sectional curvature c < -3. Now take B" to be the unit ball in C' with the metric ds2=4(1-

IziI2)(EdzZdzz)+(E 2;jdz;,)(Eztidz;,)

(1- E zzl2)2 This metric has constant holomorphic curvature -1 and the fundamental 2form 1 of the Kahler structure is given by

ndzj)+(Ezj dzj)A(Ezj dzj)

Q=dw =

(1-EIzjl2)2

d Re

4i > zjdzj

1-EIzjI2l/

Then B" x 118 with contact form rl = w + dt and metric ds2 = ds2 + ®97 is a Sasakian space form with constant 0-sectional curvature c = -4. It is well known that setting the imaginary part of zj equal to zero gives an

imbedding of real hyperbolic space H" of constant curvature -1 as a totally real, totally geodesic submanifold of B'. To construct a non-trivial example of M3 as an integral submanifold of M7(-4) = B3 x R with C-parallel second fundamental form, we consider an umbilical surface in H3 and as t varies in I[8 we rotate H3 so that the surface will trace out the desired M3. For this purpose it will be convenient to use the polar form rjeiei of zj. Then rl, and ds2 become. r?dB7

r7 = 41- F

_

r

+ dt,

a = (9tt

and

ds2

(1-E rj2)(E(dr? +r?d6, ))+(E rjdrj )2 + (1 - rj2)2

8.3 Legendre curves

r2d0j)2

133

>r2d0jdt Er7e +dt.

+16(1-Er2).72+81

We remark that the induced metric on H3 in IBa3 defined by 8; = constant is the Beltrami-Klein model and not the Poincare model of hyperbolic space. In this model an umbilical submanifold is in general an ellipsoid. With this in mind we construct our example as follows. On the Sasakian space

form M7(-4) = B3 x R we designate the coordinates by (x1, ... x7) _ ,

(rl, r2i r3, 01, B27 B3, t). Now define t : M -_> M7(-4) by

(ri, r2, t) ~' Crl r2, b

1 - ri

r2, 0, 0,

\ tJ - 1-b2 4b2 t'

where b is a constant such that b2 < 1. It is easy to see that rl(t a1) = rl(t,k82) _ rl(c*87) = 0. Thus M is a 3-dimensional integral submanifold of the Sasakian

space form M7(-4). Direct computation then shows that M has C-parallel second fundamental form (see Baikoussis, Blair, and Koufogiorgos [1995] for details).

8.3

Legendre curves

C. Baikoussis and Blair [1994] made a study of Legendre curves in 3-dimensional

contact metric manifolds and we present some of the results from that paper here. We begin with a simple proposition that will motivate the discussion. Proposition 8.2 In a 3-dimensional Sasakian manifold, the torsion of a Legendre curve which is not a geodesic is equal to 1.

Proof. Let y be a Legendre curve on a 3-dimensional Sasakian manifold parametrized by arc length. Differentiating 77(') = 0 along y we see from V x = -OX that V.yy is orthogonal to and hence that

k being the curvature. Thus the principal normal E2 = ±q5'y. Now by Theorem 6.3

V. E2 = ±( + V7 Y) The torsion, T, being the absolute value of the coefficient of , is r = +1. It is classical that a curve in Euclidean 3-space which is not a line is a plane curve if and only if its torsion vanishes. Relative to the Euclidean metric on R', the torsion of a Legendre curve is complicated, but we shall see that in the space R3 with its standard Sasakian structure (Examples 4.5.1, 6.7.1 and 7.4.2), denoted ][83(-3), a curve which is not a geodesic is a Legendre curve

134

8. Submanifolds of Kahler and Sasakian Manifolds

if and only if it starts as a Legendre curve and its torsion is equal to 1. More generally and more precisely we have the following theorem.

Theorem 8.12 For a smooth curve y in a 3-dimensional Sasakian manifold, set o = 77(7). If T = 1 and at one point a- = & = 0, then ry is a Legendre curve.

Remark. Without the hypothesis "at one point" it is possible to give a counterexample. The 3-dimensional sphere has three mutually orthogonal contact structures each of which together with the metric of constant curvature 1 forms a Sasakian structure (see Chapter 13). Now a curve which is a Legendre curve with respect to one of these need not be a Legendre curve with respect to the others, but the curve still has torsion 1. Proof. Let -y be a smooth curve in a 3-dimensional Sasakian manifold other than an integral curve of e which would, of course, be a geodesic. In fact for our argument we may suppose that y is never collinear with 1;. Then we may decompose V. y as

V 'y

_

o

=a 1

ryQ2+1

1-Q2.

Thus s; = Vl-a2 -+,32 and differentiating again we find that -)361 as T=1+ a/3 a2+,82 + Vl-l--

Now if a = 0, it is easy to see by taking the inner product of the above expression for V. y with Oy that V.yy is collinear with and in turn that y is collinear with l;, a contradiction. So we suppose a 0. Again an easy computation shows that )3 = Making this substitution for and

1- v2

.

using the hypothesis T = 1 we have the equation 2

2o-6-

0 = a 6-+

1-o2/J + a3a -

Clearly o- = 0 is a solution and o- = 0 implies a- = 0. Thus we now assume that a is non-constant. Setting A _ this equation becomes dA do-

+

2aa2 1 - Q2

Integrating gives

= a2(1 - a2)(C(1 a constant. Now suppose that at one point, a- = v = 0; then since 0, C = 1. Finally since 0-2 < 1, we have that a = 0, a contradiction. a-2

a

This property is characteristic of 3-dimensional Sasakian manifolds as we will now show.

8.3 Legendre curves

135

Theorem 8.13 If on a 3-dimensional contact metric manifold, the torsion of Legendre curves is equal to 1, then the manifold is Sasakian.

Proof. First recall that Vx _ -OX - OhX (Lemma 6.2) and that on any 3-dimensional contact metric manifold

(Vxq5)Y = g(X + hX,Y) -,q(Y)(X + hX) (Corollary 6.4). Then let y(s) be a Legendre curve. Differentiating g(y, ) = 0 0 and hence that along y we see that g(V. y, ) + g('y,

V,:, y = a + by = rcE2,

a = g(Y, ch'Y).

Thus the principal normal E2 = (al; + bcb'y). Differentiating E2 we have

V E2 =

a b k (- Za+ + g(' +h'Y,

b2

b aaOhy y+ (- Zb-a + b),yrc

_ -ray+TE3. Now h-y is orthogonal to and hence in terms of and Oy we have h-y = Si + cOy. Applying 0 we note that e = -a. Thus from our expressions for V jE2 we have

-E3=(- Za+-+-(1+8))t;+(- Zb+---(1+S))0ry. From this and K2 = a2 + b2 we obtain

T-

ba

- ab 2

+(1+S)

.

If now -y is an eigencurve of h, then e = -a = 0 along the curve and we have that T = 1 + S. Thus if every Legendre curve has torsion 1, h has zero eigenvalues everywhere and hence h = 0, completing the proof. In Example 5.3.3 we saw that the projection y* of a closed Legendre curve y in the space R3(-3) to the xy-plane must have self-intersections and moreover that the algebraic (signed) area enclosed is zero. We now mention briefly the curvature of Legendre curves in R3(-3). In the paper, Baikoussis and Blair [1994], the following result was also obtained.

Theorem 8.14 The curvature of a Legendre curve in 1183 with its standard Sasakian structure is equal to twice the curvature of its projection to the xyplane with respect to the Euclidean metric. Now a curve is a geodesic if and only if rc = 0. Thus the Legendre curves of 1R3(-3) which are geodesics are curves for which x(s) and y(s) are linear functions of s and z(s) is a quadratic function of s.

9 Tangent Bundles and Tangent Sphere Bundles

9.1

Tangent bundles

In the first two sections of this chapter we discuss the geometry of the tangent bundle and the tangent sphere bundle. In Section 3 we briefly present a more general construction on vector bundles and in Section 4 specialize to the case of the normal bundle of a submanifold. The formalism for the tangent bundle and the tangent sphere bundle is of sufficient importance to warrant its own development, rather than specializing from the vector bundle case. As we saw

in Chapter 1, the cotangent bundle of a manifold has a natural symplectic structure and we will see here that the same is true of the tangent bundle of a Riemannian manifold.

Let M be an (n+1)-dimensional C°° manifold and -r : TM -f M its tangent bundle. If (x', ... , xn+1) are local coordinates on M, set qi = x' o r; then (q', ... , qn+l) together with the fibre coordinates (v', ... , vn+1) form local coordinates on TM.

If X is a vector field on M, its vertical lift XV on TM is the vector field defined by XVw = w(X) o r where w is a 1-form on M, which on the left side of this equation is regarded as a function on TM. For an affine connection D on M, the horizontal li''t XH of X is defined by XHw = DXw. Since DXw has local expression (Xiaxi - Xiwkr )dxi, if we evaluate DXw on a vector t = v1 we have easily

(Dxw)(t) =v'X28xi

(x - 4 -XivuI' ii

kk)wlvl.

138

9. Tangent Bundles and Tangent Sphere Bundles

Thus the local expression for XH is

XH-X2a

ii-Xyv1r'avk.

4

The span of the horizontal lifts at t E TM is called the horizontal subspace of TtTM. The connection map K : TTM - ) TM is defined by

KXH = 0,

KXt = Xf,(t), t e TM.

The connection map K may also be defined in the following way. Given X E TtTM, let y be a smooth curve in TM with tangent vector X at t = y(0). Let a = -r o y be the projection of the curve to M. Then

KX = Daylo and the curve y in TM is horizontal if y, viewed as a vector field along a, is parallel. The horizontal subspace can also be defined by {(s o a),,(0) E TtTMIa path in M, s section of TM, s(a(0)) = t, D«(o)s = 0}.

It is immediate that [XV,YV]w = 0. Furthermore

[XH,YV] = [Xia

i

q

_ x1

-Xiv'I'a,YlavlI

k

axz

+ xiY`r ) avk = (DXY)v.

Similarly denoting the curvature tensor of D on M by R we have at the point

t c TM [XH,YH]t =

y]H

-

(RXYt)V.

TM admits an almost complex structure J defined by

JXH = XV, JXV = -XH. Using the above expressions for the Lie brackets in the Nijenhuis torsion of J one can easily see that J is integrable if and only if D has vanishing curvature and torsion (Hsu [1960], Dombrowski [1962]). If now G is a Riemannian metric on M and D its Levi-Civita connection, we define a Riemannian metric g on TM called the Sasaki metric, Sasaki [1958], (not to be confused with a Sasakian structure), by g(X, Y) = (G(Tr'X, Tr*Y) + G(KX, KY)) o -r

9.1 Tangent bundles

139

where X and Y are vector fields on TM. Since r* o J = -K and K o J = * is Hermitian for the almost complex structure J. On TM define a 1-form 0 by O(X), = t E TM or equivalently by the local expression o = E Gi;vidqj. Then d13 is a symplectic structure on TM and in particular 2d,3 is the fundamental 2-form of the almost Hermitian structure (J, g). To see this first note that since j3(X') = 0 and [Xv, Yv] = 0, 2d(3(Xv, Yv) = 0. Similarly 2d,6(Xv, YH)t = Xv(G(t, -r,,yH) oar) = XV (G(v1

,Y) o ar)

= G(X, Y) = g(XV,Yv) = g(XV, JYH). Now choose a vector field Z on M such that Zm, = t and (DxZ),,,, = 0 for all X. Then 2d/3(XH,YH)t = (XH(G(Z, Y)

o Tr)

- YH(G(Z, X) o Tr) - G(t, [X, Y]))m,

= G(t, DxY)m, - G(t, DyX)m, - G(t, [X,Y])m = 0.

Thus TM has an almost Kahler structure which is Kahlerian if and only if (M, G) is flat (Tachibana and Okumura [1962]). As before we let R denote the curvature tensor of D which is now the LeviCivita connection of G. The Levi-Civita connection V of g and its curvature tensor R were computed by Kowalski [1971]. These are given by the following formulas and we will give a partial proof as an illustration.

(VxHYH)t = (DxY)H - 2(Rxyt)t ,

(OxHY')t = (Rt yX)H + (DxY)c

,

2

(7xvYH)t = 1(RtxY)H VxvYv = 0.

For example from 2g(OxY, W) = Xg(Y, W) + Yg(X, W) - Wg(X, Y) + g([X,Y],W)+g([W,X],Y) -g([Y,W],X) we have 2g(VxHYv,WV) = XHg(YV,WV)+g((DxY)v,WV) -g((DxT,V)V,YV)

= g((DxY)v,WV) +G(DxY,W)

o Tr

= 2g((DxY)v,Wv)

and

2g(OxNYv, WH)t = Yvg(XH, WH) + g((Rx wt)', Yv)

140

9. Tangent Bundles and Tangent Sphere Bundles

= G(RtyX, W)

g((RtYX)H, WH)

giving the formula for (7xHYV )t. Turning to the curvature we have Rxv yv ZV = 0,

(RxvyvZH)t = (RxyZ+ 4RtxRtyZ - 4RtyRtxZ)H, (RxHyvZV)t = -(2RYzX + 4RtyRtzX)H ((DxR)tYZ) H + (2Rx zY + 4RRtyzXt) t

(RXH yvZH)t =

,

2

(RxHYHZV)t = 2((DxR)tzY- (DyR)tzX)H 1

1

v

+(RxYZ+ 4RRtzYXt - 4RRtzXYt)t

(RxH yHZH)t = (RxyZ + 4RtRzytX + 4RtRxztY + 2RtRxytZ)H

+2 ((DzR)xyt)t We will prove the fourth one of these formulas. Recall that we may write the point t as vi -a ; this is important when we have to differentiate with respect by az. to position. Also we abbreviate RXHYVZH

=

-Dyv ((DxZ)H - I vi(Rx zai) V) - V (Dxy)V ZH. Therefore

(RXHyvZH)t =

-2Xiv'r (RakyZ)H+2vi((DxRa1yZ)H- 2(RxRaiyzt)t )

-2(RtyDxZ)H + 1(RX zY)t -

1(RtDxYZ)H

from which the fourth curvature formula readily follows. The main result of Kowalski [1971] is the following theorem.

Theorem 9.1 The tangent bundle TM with the Sasaki metric g is locally symmetric if and only if the base manifold (M, G) is flat in which case (TM, is flat.

9.1 Tangent bundles

141

Proof. Clearly if the base manifold is flat, then so is (TM, g); so we have only the necessity to prove. Using the above formulas for the connection and curvature of g we have

((VWHR)XHYVZV)t = VWH(-2RYZX - 4RtyRtzX)t -R((DwX)H_ 2 (Rwxt)a) YV

ZV

RXH (2 (RtyW)H+(DwY)e ) ZV

-RXHyv(2(RtZW)H + (DWZ)t ) Using the hypothesis and taking the vertical part we obtain 0

2RW (2RYZX+4RtYRtzX)t

-(RX1RtYWZ-{-

-

(41

RRtz2RtyWXt- 1RRtzX1RtYWt)

Xt + 2RX 2RtzWY) .

In this expression set Y = t and Z = t to get respectively RW RtZXt - RX RtZWt = 0,

RWRYtXt - 2RXRtYWt = 0.

Now replace Y by Z in the second of these equations and compare with the first to obtain RX RtZWt = 0.

Setting W = X and taking the inner product with Z yields IRtZX I2 = 0 and hence that (M, G) is flat. K. Bang [1994] obtained the corresponding result for (TM, g) being conformally flat as a corollary to his result on conformally flat vector bundles, Theorem 9.11.

Theorem 9.2 The tangent bundle TM with the Sasaki metric g is conformally flat if and only if the base manifold (M, G) is flat in which case (TM, is flat.

142

9.2

9. Tangent Bundles and Tangent Sphere Bundles

Tangent sphere bundles

We have seen that principal circle bundles over symplectic manifolds form a large class of examples of contact manifolds; they have K-contact structures which are Sasakian when the base manifolds are Kahlerian. These examples together with the standard structure on R2n+1 (Examples 3.2.1, 4.5.1, 6.7.1, 7.4.2) show that Sasakian manifolds form a large and important class of contact manifolds. However, despite the example of T1 S2 = RP3 and more generally Theorem 9.3 below, the tangent and cotangent sphere bundles (Example 3.2.4) are not, in general, K-contact, even though they are classically an important class of contact manifolds. Boothby and Wang [1958] proved that a compact, simply connected, homogeneous contact manifold M of dimension

4r + 1 with r > 1 is homeomorphic to a tangent sphere bundle only when M is the Stiefel manifold V2r+2,2. In [1978] the author showed that the standard contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature cannot be regular.

We will regard the tangent sphere bundle of a Riemannian manifold as the bundle of unit tangent vectors, even though, owing to the factor 1 in the coboundary formula for d77, a homothetic change of metric will be made. (If one

adopts the convention that the a does not appear in the coboundary formula, this change is not necessary. However to be consistent the odd-dimensional sphere as a standard example of a Sasakian manifold should then be taken as a sphere of radius 2 (cf. Tashiro [1969] and Sasaki and Hatakeyama [1962]).) The tangent sphere bundle, 7r : T1M --f M, is the hypersurface of TM defined by E Gi;v'vi = 1. The vector field v = via is a unit normal as well as the position vector for a point t E T1M. We denote by g' the Riemannian metric induced on T1M from the Sasaki metric g on TM and by V its LeviCivita connection. We can easily find the Weingarten map of a hypersurface. For any vertical vector field U tangent to T1M, ;;

'UV =

Uvia' +

U.

For a horizontal tangent vector field X, we may suppose that X is the restriction of a horizontal lift; then (V(aj)HV)t = ((a )Hvi) aUi + 2vi (Rta,ej)H + v'(Da,ai)t = (Rtta,)H = 0 2

where we have again abbreviated a9; by a;. Thus the Weingarten map A of T1M with respect to the normal v is given by AU = -U for any vertical vector U and AX = 0 for any horizontal vector X. With the simple form for the Weingarten map just obtained, many computations on T1M can be done on TM. Yampol'skii [1985] (see also Borisenko

9.2 Tangent sphere bundles

143

and Yampol'skii [1987a], Tanno [1992]) computed the Levi-Civita connection and the curvature of g' but we will not need the added formalism here. We know that as a hypersurface of the almost Kahler manifold TM, T1M

inherits an almost contact metric structure. Following the usual procedure (Example 4.5.2) we define 0', ' and q' on T1M by

aH

-Jv = -viJ(8x2) = (axi)

,

JX =q'X+ 77(X)v.

(q', C', rl', g') is then an almost contact metric structure. Moreover 17' is the form on T1 M induced from the 1-form ,3 on TM, for

rl'(X) = j (z/, JX) = 2di3(v, X) = 2 E (d(Gijvj) A dqi) (vk, avk X) Gi;v'dgZ(X) =,3(X).

However g'(X, cb'Y) = 2d77'(X, Y), so strictly speaking W, ', r)', g') is not a contact metric structure. Of course the difficulty is easily rectified and 4g,

0=0', g=

7n= 271',

is taken as the standard contact metric structure on T1M. In local coordinates = 2v'(

i)H;

the vector field vi(a ;)H is the so-called geodesic flow on T1M. Before proceeding to our theorems we obtain explicitly the covariant deriva

tives of e and 0. For a horizontal tangent vector field we may again use a horizontal lift; then (XH2v2)(0i)H + 2vi(Dxai)H

- (Rxtt)i = -(Rxtt)t

and hence for any horizontal vector X we have

(Vxe)t = -(R,.xtt)i For a vertical vector field U we have

(VuC)t = (ous)t = (U2vi)(ai)H - vi(RKUt(ai))H = -2(gU)t - (RKUtt)H,

since (ei)H = -J(ai)v. Comparing with Vxe = -OX - qhX, we have for X horizontal and orthogonal to

and for U vertical

hXt = -Xt + (R,,xtt)H,

hUt = Ut

- (RKUtt)c

144

9. Tangent Bundles and Tangent Sphere Bundles

To differentiate 0, first note that for any tangent vector fields X and Y,

(Vxo)Y =1IxJY - (Vxrj)(Y)v + q(Y)AX -g'(X, AOY)v - J17xY - g'(X, AY)E'. Again for X, Y horizontal we suppose that they are horizontal lifts and we let U = U' '9 - and W = W' av be vertical vector fields tangent to TIM. Also denote by "tan" the component of a vector in TtTM, t E TiM that is tangent to TIM. Then recalling that AX = 0 for X horizontal we have

-2 (R,.Yt7.X)H + (D,,.x.7r*Y)t - (Vx?1') (Y)v 1

-J(D,.x7r*Y)H + J(R,,.x,,.Yt)t 2

= 2 where we have used the first Bianchi identity and our formula above for Vx6.

((Vx0)U)t = -(XUz)(a,)H - vi(D,.x(9i)H+ 2(RT.xxut)t -

-J(XU')(ai)t

U'J(D,,.xai)t + 2J(Rxut7r*X)H = 2 tan(R,r.xtKU)t

where we have used (Vx7')(U) = 29((R, xtKU)t,v). In a similar manner one obtains ((VUO)X)t = -217(X)Ut+ 2 tan(Rxut7*X)t ((Vuo)W)t =

2(RxutKW)H.

We now prove a theorem of Tashiro [1969] which shows that the contact metric structure on the tangent sphere bundle is almost never Sasakian.

Theorem 9.3 The contact metric structure (0, , 71, g) on TIM is K-contact if and only if the base manifold (M, G) has positive constant curvature +1 in which case TIM is Sasakian.

9.2 Tangent sphere bundles

145

Proof. If l;, rl, g) is a K-contact structure, then Vx = -OX as we have seen, but for a horizontal lift we have (VXHe)t = -(Rxtt)' and hence (¢XH)t = (Rxtt)''. Now for X orthogonal to t, OXH = Xv and therefore Rxtt = X for all orthogonal pairs {X, t} on (M, G) from which we have RxyZ = G(Y, Z)X - G(X, Z)Y. Conversely by the formulas above for the covariant derivative of 0, the condition RxYZ = G(Y, Z)X - G(X, Z)Y on (M, G) implies ((Vxo)Y)t = - I (G(7r*Y, t)7r*X - G(7r,,X, 7r*Y)t)H = (g(X, Y) - rl(Y)X )t,

((VXO)U)t = 2 tan(G(KU, t)7r*X - G(7r*X, KU)t)t = 0,

((VUO)X)t = -277(X)U + 2 tan(G(t, 7r*X)KU - G(KU, 7rX)t)t = -rl(X)Ut and

((VUO)W)t = 2g(U, W) + 2 (G(t, KW)KU - G(KU, KW)t)H = g(U W) showing that T1M is Sasakian.

In particular if the base manifold is S3(1), then the standard contact metric structure on T1S3(1) - S3 X S2 is Sasakian. Moreover this metric is known to be Einstein but it is not a Riemannian product of constant curvature metrics. S3 X S2 is therefore a nice example of a manifold with more than one Einstein

metric, (see e.g., Tanno [1978b]). If go denotes the standard metric on the sphere, (S3, 2go) x (S2, go) is Einstein but clearly not an associated metric for any contact structure in view of Pastore's result [1998] on 5-dimensional locally symmetric contact metric manifolds. Recall the symmetric operator 1 (Section 7.2) which was defined by 1X = Rxe For a K-contact structure, 1 = I - ri ® and there exist many contact metric manifolds for which 1 = 0, as we will see in Section 4, Theorem 9.14. .

More strongly we have seen that Rx y = 0 implies that the contact metric manifold M2"+1 is locally isometric to En+1 X Sn(4) (Theorem 7.5) which is the tangent sphere bundle of Euclidean space. In the case of the contact metric structure on T1M we have the following result (Blair [1977]).

Theorem 9.4 The standard contact metric structure (0,

g) on T1M satisfies l U = 0 for vertical vector fields U if and only if the base manifold (M, G)

is flat in which case we have Rxye = 0 for all X and Y on T1M. Proof. First note that for a vertical vector field U tangent to T1M Ru implies Ru£ = 0. Now recall that

K(RxH yvZH)t = 2Rx zY + 4RRt,,zxt.

=0

146

9. Tangent Bundles and Tangent Sphere Bundles

Since

= 2vi(ay )H, combining these facts gives RRKU,ttt = 0

i.e., RRYXxXX = 0 for any orthonormal pair on (M, G). Taking the inner product with Y gives IRyxXI2 = 0 from which we see that (M, G) is flat. Conversely if (M, G) is flat, the Gauss equation for T1M as a hypersurface of TM gives g'(R'xye, Z) = 0 and hence that Rxye = 0. If the base manifold (M, G) has constant curvature -1, then the sectional curvature K(U, Z;) = 1 for any vertical vector U tangent to T1M and K(X,

-7 for any horizontal vector tangent to T1M. In Section 7.2 we remarked that the classification of all locally symmetric contact metric manifolds remains open. Here we will show that the contact metric structure on T1M is locally symmetric if and only if either the base manifold is flat or 2-dimensional and of constant curvature +1 (Blair [1989]). Here to do this we first give a couple of lemmas and then utilize a theorem of D. Perrone [1994].

Lemma 9.1 If a contact metric manifold is locally symmetric, V 6h = 0.

Proof. Since the metric is locally symmetric and the integral curves of are geodesics, we see that the operator 1 is parallel along 1. Thus since V = 0 for any contact metric structure, differentiation of the second equation of Proposition 7.1 gives V h2 = 0. Then differentiating the first equation of Proposition 7.1 we have VeV£h = 0. WritingVeh2 = 0 as (VEh)h+h(V h) = 0 and differentiating we have (V h)2 = 0. Since V eh is a symmetric operator, we have the result. For each unit tangent vector t E T,,,,M, let [t]1 be the subspace of TmM orthogonal to t and define a symmetric linear transformation Lt : [t]-L -> [t]1

by LtX = Rxtt. Lemma 9.2 If the contact metric structure on T1M satisfies V h = 0, then for any orthonormal pair {X, t} on (M, G), LtX = LtX and (M, G) is locally symmetric.

Proof. Since hUt = Ut - (RKUtt)V for a vertical vector U E TtT1M we have h2Ut = Ut - 2(LtKU)v + (Lt KU)'. Proceeding as in the proof of Theorem 9.4 we also have lUt = (Lt KU)'' + 2((DtR)KUtt)H. On the other hand since DEh = 0, applying 0 to the first equation of Proposition 7.1 gives ¢12 + h2 +I = 0. Applying this to U, the vertical part gives Lt = Lt on (M, G). The horizontal part gives us that (M, G) is locally symmetric by a result of Cartan that a Riemannian manifold (M, G) is locally symmetric if and only if G((DxR)y xY, X) = 0 for all orthonormal pairs {X, Y} (English translation: Cartan [1983, pp. 257-258]).

9.2 Tangent sphere bundles

147

Theorem 9.5 The standard contact metric structure (0,

g) on T1M satisfies V h = 0 if and only if the base manifold (M, G) is of constant curvature 0 or +1. Proof. This theorem follows from Lemma 9.2 and the purely Riemannian result (Perrone [1994]) that a Riemannian manifold (M, G) is locally symmetric and satisfies Lt = Lt if and only if (M, G) is of constant curvature 0 or +1.

Theorem 9.6 The standard contact metric structure (0, , 71, g) on T1M is locally symmetric if and only if either the base manifold (M, G) is flat or 2-dimensional and of constant curvature +1.

Proof. By Lemma 9.1 and Theorem 9.5 we see that if the structure T1 M is locally symmetric, then (M, G) is of constant curvature 0 or 1. If (M, G) has constant curvature 1, then T1M is Sasakian by Theorem 9.3 and therefore of constant curvature +1 by Theorem 7.11. That dim M is now 2 follows from Musso-Tricerri [1988] who prove that (T1M, g') is Einstein only if dim M = 2 or by Theorem 9.7 below on the conformally flat case. It remains to prove the converse, that if (M, C) is flat or of dimension 2 and constant curvature 1, then (T1M, g) is locally symmetric. In the flat case this is virtually evident for a number of reasons. For example, (TM, g) is flat and is horizontal and so on (T1M, g) we have RXY = RXyI = 0. Then by Theorem 7.5, (T1M, g) is locally isometric to E'+1 x S'(4) for n > 1 and flat for n = 1. For (M, G) of dimension 2 and constant curvature 1, a direct computation shows that (T1M, g') has constant curvature 1 and therefore (T1M, g) has constant curvature 1. The corresponding result for (TIM, g) being conformally flat is stronger (Blair and Koufogiorgos [1994]).

Theorem 9.7 The standard contact metric structure on T1M is conformally at if and only if the base manifold (M, G) is a surface constant Gaussian curvature 0 or +1. In Section 6.4 we saw that a contact metric structure gives rise to a strongly pseudo-convex CR-structure if and only if

(Vxq)Y = g(X + hX, Y) -,q(Y) (X + hX). For the tangent sphere bundle we have the following result of Mitric [1991], Tanno [1992].

Theorem 9.8 For dim M > 3 the standard contact metric structure on T1M gives rise to a strongly pseudo-convex CR-structure if and only if the base manifold (M, G) is of constant curvature.

148

9. Tangent Bundles and Tangent Sphere Bundles

Proof. Using hUt = Ut - (RKutt)v evaluate the right-hand side of

(VXq)Y = g(X + hX,Y) - 77(Y)(X + hX) on two vertical vectors U and W and compare with our earlier expression for the left-hand side. The right-hand side becomes 2g(U, while the left-hand side is 2g(U, W)l;t + (RKutKW)H. Equality implies that a with t for all X, Y orthogonal to on the base manifold, RX tY is collinear t. In particular G(RX tY, X) = 0 for every orthonormal triple It, X, Y} and hence (M, G) is of constant curvature. Conversely if RX yZ = c(G(Y, Z)X (Y)(X+hX) G(X, Z)Y), evaluate both sides of (Vxq)Y = g(X +hX, for the four cases of X, Y being both horizontal, both vertical, and each one horizontal with the other vertical and compare. Tanno actually proves more; in [1992] (and [1991]) he introduces a gauge invariant B of type (1,3) (on the contact subbundle) whose vanishing implies the CR-condition of Theorem 6.6 and conversely under the CR-condition B reduces to the Chern-Moser-Tanaka invariant. See Section 10.5 for the notion of a gauge transformation. Tanno proves that on T1M, dim M > 3, B vanishes if and only if (M, G) is of constant curvature -1. Th. Koufogiorgos studied in [1997a] the idea of constant cl-sectional curvature for non-Sasakian contact metric manifolds, especially for (ic, p)-manifolds. For the tangent sphere bundle he proved the following theorem.

Theorem 9.9 If (M, G) is of constant curvature c and dimension > 3, the standard contact metric structure on T1M has constant cl-sectional curvature (equal to c2) if and only if c = 2 ± -,/5-. 1, (T1M, g) has constant For (M, G) a surface of constant curvature c sectional curvature c2 as follows readily from Theorem 7.9 and the formula K(X, OX) = -(s; + p) on the same page.

9.3

Geometry of vector bundles

The geometric constructions on the tangent bundle described in Section 9.1 can be carried out on a general vector bundle. In this section we describe this construction and the corresponding results without proofs. We follow the treatment given by K. Bang in [1994]. Let (M', G) be a Riemannian manifold with Levi-Civita connection D and curvature tensor R as before. Consider a vector bundle it : En+k --3 Mn with fibre metric g1 and a metric connection V. If (x1, ... , xn) are local coordinates on M, set qi = xi o it; if {e} is a local orthonormal basis of sections of E, writing a point (m, U) E E as U = > u"ea (q1 qn) together with the fibre coordinates (u1, ... , uk) form local coordinates on E. ,

9.3 Geometry of vector bundles

For a section

_

149

(«e« of E, the connection V is given by

a«

l

Xz (\8xi

Vx( =

where Vaiep = If X is a tangent vector field on M and if C is a section of E, the horizontal lift XH of X and the vertical lift (V of < are defined by XH = XZ

84

ii - Xiu'°µaza8

and

tV =,«a« Then 7r*XH = X and it*(V = 0. Define a linear map K : TE - -) E by KXH = 0,

U) E E

K((m,U)

or by its local expression: If (Xi, X'+«) are the components of a vector X tangent to E at (m, U) with respect to the coordinate basis, then

KX = (Xn+-Xiuaµ)3i)e« We define a Riemannian metric g on E, called the Sasaki metric, by

g(X, k) = G(7r*X, 7r*Y) + gl(KX, KY)

where X and k are vector fields on E. When E = TM and 17 = D, g is the Sasaki metric on TM.

The curvature tensor R of V is given by Rxyc = VXVYC - VyVxc V [x,Y]C and V is said to be flat if R vanishes for all vector fields X, Y on M

and all sections ( of E. Since Rx y is also a section of E, we can compute its inner product gl(Rxy(,,O) with another section 0. We then define the adjoint &OX by G(R( ,/X, Y) = g1(Rxy(, 0) With this notation in mind we give the the covariant derivatives of the LeviCivita connection V of the Sasaki metric g at a point (m, U) E E. VXHYH

= (DxY)H - 2(RxyU)v, ,,7SvYH

VxH(V

= 2(RUSY)H,

= 2(RUSX)H + (Vx()v,

7C, V = 0.

150

9. Tangent Bundles and Tangent Sphere Bundles

The curvature R of the Sasaki metric at (m, U) is given as follows. R vanishes on three vertical lifts and R(vOvZH = (Rc,yz+ 4RucRuOZ - 4 RUORU(Z)H,

RXHcv'ov = -(2R(OX + 4-Ru(RU X)H, RXH(vZH=

1RRUczxU)V,

((DxR)ucY-(DyR)U(X)H

R XHYH(V 1

+(Rxy( + 4RRUcyxU

1

4RfzucxYU)

v

4URXZUY+ RXHYHZH = (RxYZ+ 4RuR2,,UX +R

+2

2RURXYUZ)x

((VzR)xYU)v.

Concerning the questions of the vector bundle E with the Sasaki metric being locally symmetric or conformally flat, K. Bang proved the following theorems.

Mn be a vector bundle over a Riemannian Theorem 9.10 Let 7r : En+k manifold (M, G) with fibre metric gl and a metric connection V. Then the Sasaki metric on E is locally symmetric if and only if the connection V is flat and (M, G) is locally symmetric.

Theorem 9.11 Let it : En+k , Mn be a vector bundle over a Riemannian manifold (M, G) with fibre metric gl and a metric connection V. Then the Sasaki metric on E is conformally flat if and only if either (M, G) is fiat with flat connection V, or (M, G) has constant curvature with flat connection V and k = 1. As a corollary K. Bang pointed out that the tangent bundle TM with the Sasaki metric g is conformally flat if and only if the base manifold is flat (Theorem 9.2).

9.4

Normal bundles

For the case of the normal bundle of a submanifold of a Riemannian manifold, the above construction of the Sasaki metric was developed by Reckziegel [1979]

and Borisenko and Yampol'skii [1987b] using the normal connection, i.e.,

9.4 Normal bundles

151

V = V1. In this section we consider the special cases when the submanifold is a Lagrangian submanifold of a Kahler manifold or an integral submanifold of a Sasakian manifold. Again the results were obtained by K. Bang in his thesis [1994].

First let L be a Lagrangian submanifold of a symplectic manifold M2n with

associated metric g and corresponding almost complex structure J. Recall that if X is tangent to L, then JX is normal. Typically normal vectors will be denoted by (, v, 0. Using the ideas of horizontal and vertical lifts in the case of vector bundles from the last section, we define an almost complex structure

J on the normal bundle T-L of L by

JXH = (JX)V,

jcV

= (JO)H.

That J2 = -I and that the Sasaki metric g is Hermitian with respect to J are easily verified. However the symplectic nature of (T±L, j, g) depends on the ambient manifold (M2n, j, g) being Kahler. Let 1 be the fundamental 2-form of the almost Hermitian structure just defined on T-'-L.

Theorem 9.12 Let L be a Lagrangian submanifold of a Kahler manifold (M2n, j, g). Then the normal bundle (T'L, SZ) is a symplectic manifold.

Proof. Since (J, g) is an almost Hermitian structure, it is immediate that Stn

0, so we have only to show that SZ is a closed 2-form. First it is immediate

that S2(XH,YH) = 0, S1(XH,(V) = g(X,J(), and n((V,iV) = 0. Now for horizontal lifts of vector fields tangent to L, we have [XH,YH] = [X, y]H - (RXYV)V at the point v E T1L. Thus using the coboundary formula for dSZ we have 3dcl(XH,YH, ZH)

= XHI (YH, ZH) + YHSZ(ZH, XH) + ZHSZ(XH,YH) -0([XH,YH],ZH) n([YH,ZH],XH) n([ZH,XH],YH)

-

=

-

!n((RXYV)V, ZH) + SZ((RYZV)V, XH) + n((RZXv)V,YH).

In keeping with the notation of this chapter, let G be the induced metric on L, D its Levi-Civita connection and R its curvature tensor. Using the GaussWeingarten equations and the Kahler condition, one readily obtains JVXV = DXJv. Thus we have !n ((RxYV)V , ZH) = ((RXYV)v, JZH) _ -g((JRxYV)H, ZH)

152

9. Tangent Bundles and Tangent Sphere Bundles

= -G(RxyJv, Z) = G(RXYZ, Jv). Substituting this and like expressions in the coboundary formula and using the Bianchi identity we have 3dQ(XH, yH, ZH) = 0. In much the same way one shows that dQ(XH yH (V) = 0 and dQ(XH, (V, O V) = 0, and of course df vanishes on three vertical vectors. We now turn to the question of when the above almost Kahler structure on T'L is itself Kahler.

Theorem 9.13 Let L be a Lagrangian submanifold of a Kahler manifold (M2n, j, g). Then the following are equivalent: i) (T'L, J, g) is Kahler. ii) L has flat normal connection. iii) L is flat.

Proof. In the previous proof we noted that JVXV = DXJv and therefore RxyJv = JRXYv. Thus L is flat if and only if L has flat normal connection. To complete the proof we show that the almost complex structure J on T'L is integrable if and only if L has flat normal connection. This will be done by computing the Nijenhuis torsion of J. [J,

(XH, (V) _ -[XH, (V] + [(JX )V , (J()H]

- J[(JX)v, (V] - J[XH, (J(H]

J([X, J(]H - (RX,7(v)V) _ -(Vx()V _ -(VX(+V-SJX + J[X, J(])V + (JRX J(v)H.

Now using the Kahler condition and the fact that J[X, J(] is normal to L one J[X, J(] = 0. Therefore can show that VX( + V [J, J](XH, (v) = (JRXJSv)H. Similarly [J, J] (X H, YH) = (R4YV)V,

[J, J] ((V >V) _ -(R,S

v)V

and the result follows.

We remark that in Chapter 1 we proved a theorem of Weinstein that a symplectic manifold is locally the cotangent bundle of any Lagrangian submanifold. Here one may note that, since JVXV = DXJv, j provides a connection preserving isomorphism between the tangent bundle and the normal bundle of L. Turning to the contact case let M" be an integral submanifold of a contact metric manifold M1r+1 with structure tensors (0, , i , g). On the normal

bundle T'M", we define an almost contact structure (0, , ) by

XH = (4)v,

ff

= 0,

RV = (k)H

9.4 Normal bundles

for all tangent vectors X and normal vectors c orthogonal to

(X) = 9(X,

V,

.

153

Also let

)

1 and 2 = -I + 77 ® follows easily. for any vector X. Then Using the coboundary formula for drl we have at a point v E TIM-,

-YH_(XH) - i ([XH,YH])

_ -9([XH,YH],) = 9(RxYV,) = 9(RxYv,) +9([A,,,Ag]X,Y) by the equation of Ricci-Kiihn. For a normal vector ( orthogonal to we have 2di (XH,

CV)

= -9([XH, (v],

-9(V4(,

-g(tx(,

fix) = 0, di7(Cv, ) = 0 and di ((v, Ov) = 0. Similarly If now M2n+1 is Sasakian, by Lemma 8.1, A = 0 and, by Proposition 7.3, fixy _ (Y)X -i7(X)Y, giving 0. Thus RXYe = 0 for any tangent vectors X and Y and in turn &,X = 0 for all v E TIM'. Now setting 2

77= 277,

,

9= 49,

we see that d17(X, Y) = g(X, OY) for all vector fields on T1M giving T-Mn a contact metric structure (0, , rl, g). In Section 7.2 we mentioned in passing that there exist many contact metric manifolds satisfying l = 0 (IX = This is seen from the following theorem of Bang [1994].

Theorem 9.14 Let Mn be an integral submanifold of a Sasakian manifold g). Then the normal bundle, T-Mn, has Men+1 with structure tensors a contact metric structure (0, , rj, g) satisfying l = 0. Proof. We have just seen that when Mn is an integral submanifold of a Sasakian manifold Men+1, T'Mn has a contact metric structure (0, , 97, g). So it remains only to show that l = 0. From the curvature expressions of Section 9.3 and

0 we have RxHg

V

X)H=0

and RSv g = 0.

In particular we see that the contact metric structure (0, e, rl, g) on T'Mn is never Sasakian, so we have only a partial Sasakian analogue of Theorem 9.12. This is the following result of Yano and Kon [1983, p. 50].

154

9. Tangent Bundles and Tangent Sphere Bundles

Theorem 9.15 Let M' be an integral submanifold of a Sasakian manifold M2n+1. Then Mn has flat normal connection if and only if (Mn, G) has stant curvature 1.

con-

As an example of Theorem 9.15 recall Example 5.3.1 of Sn as a totally In this case geodesic integral submanifold of the Sasakian manifold the normal bundle T1Sn has flat normal connection and T'Sn as a contact metric manifold with the structure rl, g) is again the common example S2n+l.

En+1 X Sn(4) (cf. Theorem 7.5). We also point out that for a normal vector C orthogonal to l;, we have on 0 where V is the Levi-Civita connection of g. T1Mn that V ve =

Thus from Lemma 6.2, h(v = -rV and hence -1 is an eigenvalue of h with multiplicity n. Since ho + Oh = 0, +1 is also an eigenvalue with multiplicity n and hXH = XH. From [XH,YH] = [X,Y]H - (RXYV)V we see that the subbundle [+1] is integrable if and only if Mn has flat normal connection. We close this chapter with a continuation of Example 5.3.2. There we gave an imbedding of the 2-torus as a flat minimal integral submanifold of S5. In keeping with the notation of Section 9.3 we let {x', x2 } be the local coordinates

on T2 instead of {u, v}; Xl = a-l , X2 = a-' are orthonormal in the metric G, the restriction of g to T. Let el = cpX1, e2 = X2 and e3 = . To find Va ep = µ'Zea explicitly, compute Vea using the Sasakian condition. Then one finds 1

N'3

1--1,

c32/-ill = 1, 2

3

3

/'22=1

as the non-zero M %'s. Also {ul, u2, u3} denote the fibre coordinates so that {qi = x o 7r, u"} are local coordinates on T--T2. Now computing the Sasaki metric g we have a

g

g

a1

ai>ag;J

a a aqi , auc,

=

+ a

lj1piµa; upub

1

a

s I

auo', aua I =

Sip.

Thus for the contact metric metric structure (rl, g) on T'T2 we have g = 2 (du3 + uldgl + u2dq2) and 1 + (ul)2 + (u3)2

ulu2

ulu2

1 + (u2)2 + (u3)2

-u3 0

0

ul

-u3 u2

-u3

0

1

0

0

0

-u3

0

1

0

ul

u2

0

0

1

9.4 Normal bundles

155

Compare this metric with the metric 1

g=4

b, + yiyj + b'jz2 6 z

-yi

8i; z

bi;

0

-y,

0

1

associated to the Darboux form = (dz - E yidxi) on j2n+1 introduced 2 of the flat associated metric of the in Section 7.2 as a formal generalization 3 Darboux form on 1R.

10 Curvature Functionals on Spaces of Associated Metrics

10.1

Introduction to critical metric problems

The study of the integral of the scalar curvature, A(g) = fm T dV9, as a functional on the set M1 of all Riemannian metrics of the same total volume on a compact orientable manifold M is now classical, dating back to Hilbert [1915] (see also Nagano [1967]). A Riemannian metric g is a critical point of A(g) if and only if g is an Einstein metric. Since there are so many Riemannian metrics on a manifold, one can regard, philosophically, the finding of critical metrics as an approach to searching for the best metric for the given manifold. Other functions of the curvature have been taken as integrands as well, most notably B(g) = fm T2 dV9, C(g) = fm IpI2 dVg where p is the Ricci tensor, and D(g) = fm IRkjihl2 dVg; the critical point conditions for these have been computed by Berger [1970]. From the critical point conditions it is easy to

see that Einstein metrics are critical for B(g) and C(g) but not necessarily conversely. For example an q-Einstein manifold M1r`+1 with scalar curvature

equal to 2n(2n + 1) or 2n(2n + 3) is a non-Einstein critical metric of C(g), Yamaguchi and Chuman [1983]. In the case of B(g) Yamaguchi and Chuman showed that a Sasakian critical point is Einstein. Similarly metrics of constant curvature and Kahler metrics of constant holomorphic curvature are critical for D(g), see Muto [1975]; also a Sasakian manifold of dimension m and con-

stant 0-sectional curvature 3m - 1 is critical for D(g), see Yamaguchi and Chuman [1983]. Our study in this chapter is primarily motivated by two kinds of questions.

1) Given an integral functional restricted to a smaller set of metrics, what is the critical point condition; one would expect a weaker one. The smaller sets

158

10. Curvature Functionals on Spaces of Associated Metrics

of metrics we have in mind are the sets of metrics associated to a symplectic or contact structure. 2) Given these sets of metrics, are there other natural integrands depending on the structure as well as the curvature? To introduce the techniques for our study we will prove the classical result that a Riemannian metric is critical for A(g) if and only if it is Einstein. Let M be a compact orientable manifold and M1 the set of all Riemannian metrics normalized by the condition of having the same total volume, usually taken to be 1, but one need not insist on the particular value in a given problem. As in Section 4.3 we often denote by the same letter a tensor field of type (0,2) and its corresponding types (1,1) and (2,0), determined by the metric under consideration, e.g., we may write trTD = TijDji = T'jDi;.

Lemma 10.1 Let T be a second order symmetric tensor field on M. Then fm trTD dVg = 0 for all symmetric tensor fields D satisfying fm trD dVg = 0 if and only if T = cg for some constant c.

Proof. If T = cg, trTD = ctrD and the sufficiency is immediate. Thus we have only to prove the necessity. Let X, Y be an orthonormal pair of vector fields on a neighborhood U on M and f a C°° function with compact support in U. Regarding X and Y as part of a local orthonormal basis, define a tensor field D on M by D (X, X) = f and D (Y, Y) = - f , with all other components equal to zero and D = 0 outside U. Then fm (T (X, X) - T (Y, Y)) f dVg = 0 for any C°° function with compact support and hence T (X, X) = T (Y, Y) for every orthonormal pair X, Y. Therefore T = cg for some function c and it remains to show that c is a constant. To see this let X be any vector field and D = £xg. Then since the integral of a divergence vanishes,

0=

IT'j('7iXj + V;Xi) dVg =

-2 fM(ViT'.j)X; dVg,

M

but X is arbitrary so that ViTij = 0 (Lemma 4.7) from which we see that c must be a constant. Now the approach to these critical point problems is to differentiate the functional in question along a path of metrics. The curvature functionals we study are not generally invariant under homothetic transformations; so when necessary we normalize these problems by restricting them to M1. Let g(t) be a path of metrics in M1 and

Di; =

agZ' at

t=O

its tangent vector at g = g(0). We define two other tensor fields by

10.1 Introduction to critical metric problems

159

Djih = 2 (VjDih + ViDjh - vhDji), Dkjih = VkDjih - VjDkih

where V denotes the Levi-Civita connection of g(0) and we note that Djih =

ar ..h

aRkjih

ate

at

t=o

t=o

where rjih and Rkji h denote the Christoffel symbols and curvature tensor of g(t)Theorem 10.1 Let M be a compact orientable Co" manifold and .M1 the set of all Riemannian metrics on M with unit volume. Then g E M1 is a critical point of A(g) = fm T dV9 if and only if g is Einstein.

Proof. The proof is to compute at at t = 0 for a path g(t) in M1. First note that from gijgjk = Sk, at t=o Differentiation of the volume element gives dtdV9

= d det(g(t))dxl A .. A dx" =

2det(g(t))

29ij (gij) dV9 =

(d

det(g(t)))dV9

2D?dV9.

Now dA

d

dt t=0

1. (Dkjik9

I

dt M

Rkjik

gjidV9 t=0

- PjiD'z + 2Tg"Dji) dV9

J (-pjZ + Tgj')Dji 2 dVg M

since the integral of a divergence vanishes. On the other hand differentiation of fm dV9 = 1 gives fm D? dV9 = 0. Thus setting at t=o = 0 and applying the lemma, we have 1

Pji - 2T9ji = cgji for some constant c and hence that g is Einstein. The converse is immedia

In [1974a] Y. Muto computed the second derivative of A(g) at a critical point and proved the following theorem.

160

10. Curvature Functionals on Spaces of Associated Metrics

Theorem 10.2 The index of A(g) and the index of -A(g) are both positive at each critical point.

Y. Muto also considered the second derivative of the functional D(g) from the following point of view. Let Di f f denote the diffeomorphism group of M; if f E Di f f , then D (f *g) = D (g) and hence we have an induced mapping

D :.M1/Di f f --> R. We say that a metric g is a critical point of D if its orbit under Di f f is a critical point of D. Recall from the introduction to this section that a Riemannian metric of constant curvature is a critical point of D(g); Y. Muto [1974b] proved the following result.

Theorem 10.3 If M is diffeomorphic to a sphere and go is a metric of positive constant curvature, then the index of D(g) and the index of b are both zero at go and b has a local minimum at go. We now turn to integral functionals defined on the set of metrics associated to a symplectic or contact structure. To begin we recall from Chapter 4 how the set A of associated metrics sits in the set N of all Riemannian metrics with the same volume element. In particular we saw that a symmetric tensor field D is tangent to a path in A at g if and only if

DJ+JD=0 in the symplectic case and

Dql+clD=0 in the contact case. Similar to the role played by Lemma 10.1 we have the following lemma for critical point problems on A.

Lemma 10.2 Let T be a second order symmetric tensor field on M. Then fMT"Dij dV = 0 for all symmetric tensor fields D satisfying DJ+ JD = 0 in the symplectic case and Dt; = 0, Do + OD = 0 in the contact case if and only if TJ = JT in the symplectic case and OT -To = rl®OTt; - (rloTO) ®1; in the contact case (i.e., 0 and T commute when restricted to the contact subbundle). Proof. We give the proof in the symplectic case; the proof in the contact case being similar (see Blair and Ledger [1986]). Let X1i... , X2n be a local J-basis defined on a neighborhood U (i.e., X1i... , X2n is an orthonormal basis with respect to g and X2ti = JX2i_1) and note that the first vector field X1 may be any unit vector field on U. Let f be a C°° function with compact support in U and define a path of metrics g(t) as follows. Make no change in g outside

U and within U change g only in the planes spanned by X1 and X2 by the matrix

1+tf + 2t2f2 2t2f2

2t2f2

1

1-tf+Zt2f2 J

10.1 Introduction to critical metric problems

161

It is easy to check that g(t) E A and clearly the only non-zero components of D are D11 = -D22 = f . Then fm TijDij dV = 0 becomes (

T'1-T22)fdV=0.

IM

Thus since X1 was any unit vector field on U,

T(X, X) = T(JX, JX) for any vector field X. Since T is symmetric, linearization gives T J = JT. Conversely, if T commutes with J and D anti-commutes with J, then trTD =

trTJDJ = trJTDJ = -trTD, giving TijDij = 0. We end this section with our first main result (Blair and Ianus [1986]), namely we consider the functional A(g) restricted to the set A and find the critical point condition. Since A is a smaller set of metrics than M1, we expect

a weaker critical point condition than being Einstein; we will see that the condition is that the Ricci operator commute with the corresponding almost complex structure and hence is still a very natural condition.

Theorem 10.4 Let M be a compact symplectic manifold and A the set of metrics associated to the symplectic form. Then g E A is a critical point of A(g) = fMT dVg restricted to A if and only if the Ricci operator Q of g commutes with the almost complex structure corresponding to g.

Proof. The proof is again to compute dA at t = 0 for a path g(t) in A. Since all associated metrics have the same volume element this is easier than in the Riemannian case. In particular we have, dA dt

=

t=0

d = dt

Rkj kgjddVg t=0

M

-

fM

dVg

f piiDj2 dVg, M

the other terms being divergences and hence contributing nothing to the integral. Setting dA It=O = 0, the result follows from Lemma 10.2.

The commutativity QJ = JQ is equivalent to p(JX, JY) = p(X, Y) and is often referred to as the J-invariance of the Ricci tensor or as the manifold having Hermitian Ricci tensor.

162

10.2

10. Curvature Functionals on Spaces of Associated Metrics

The *-scalar curvature

In Section 7.2 we defined the *-Ricci tensor and the *-scalar curvature in contact geometry. In almost Hermitian geometry these are defined similarly by *

kl

P* = Riklt J J,

t

T

*i

Pi

On a Kahler manifold, p = pij. The most important property of r* on an almost Kahler manifold is the analogue or forerunner of Proposition 7.7, viz.,

1 VJI2. Therefore r - T* < 0 with equality holding if and only if the metric is Kahler. Thus for M compact, Kahler metrics are maxima of the functional

K(g)_JMT --r*dV on A and hence it is natural to ask for the critical point condition in general. This was the main question of Blair and lanus in [1986]; the critical point condition for K(g) turns out to be the same as for A(g) on A, viz., QJ = JQ.

Theorem 10.5 Let M be a compact symplectic manifold and A the set of metrics associated to the symplectic form. Then g E A is a critical point of K(g) if and only if QJ = JQ. At first it may seem surprising that A(g) and K(g) have the same critical point condition but we will see in the course of our discussion that this is natural. The proof of Theorem 10.5 will then be an easy consequence of Theorem 10.4 and Theorem 10.6 below. It is also natural to ask whether Kdhler metrics are the only critical points of K(g); the answer to this is negative and a counterexample was given by Davidov and Muskarov [1990] on the twistor space of a compact Einstein, self-dual 4-manifold with negative scalar curvature.

In [1969] S. I. Goldberg showed that if RxyJZ = JRXyZ on an almost Kahler manifold, then the metric is Kahler, and conjectured that a compact almost Kahler-Einstein manifold is Kdhler. This conjecture is still open, but under the additional assumption of non-negative scalar curvature it was proved by Sekigawa [1987]. Without the assumption of compactness the conjecture is false; non-existence was shown by Armstrong [1998], Nurowski and Przanowski [1999] gave an example of a 4-dimensional non-Kahler almost Kahler which is Ricci flat, and Apostolov, Draghici and Moroianu [2001] have given non-

compact counterexamples to the conjecture in dimensions > 6. The scalar curvature in the twistor space example, mentioned above, of a non-Kahler, almost Kahler manifold satisfying QJ = JQ is negative; thus one might ask in

10.2 The *-scalar curvature

163

light of Sekigawa's result, if a compact almost Kahler manifold with QJ = JQ and non-negative scalar curvature is Kahler. Draghici [1999] answered this question affirmatively in dimension 4 and negatively in general dimension. As a partial result in general dimension, Draghici proved in [1994] that if M is a compact almost Kahler manifold with Hermitian Ricci tensor and if there exists a constant .\ > 0 such that \ < Ric(X) < 2A for any direction X, then M is Kahler.

The original proof of Theorem 10.5 was to proceed as in Theorem 10.4 and differentiate -T*. This is very complicated and only after clever use of many identities does one see that differentiation of this term again yields a contribution of -Pij Dij to the integrand and hence that one has the same critical point condition (precisely 2Q commuting with J). This suggests that if we consider the "total scalar curvature", I(g) = fMT+T* dV, the contributions of each term to the derivative of the integrand would have canceled each other and hence every metric in A would have been a critical point; thus, since A is path connected, I(g) must be constant on A and hence a symplectic invariant (Blair [1991a]). We now prove this using only relatively short computations.

The motivation for studying the integral of the sum of T and r* lies in the author's work with D. Perrone [1992] on critical point problems in the contact case and will be discussed later in this section.

Theorem 10.6 Let M be a compact symplectic manifold. Then fm -r +,r* dV is a symplectic invariant and to within a constant is the cup product

(ci(M)

U

[l]"-1)([M])

where cl (M) is the first Chern class of M.

Proof. Consider an almost Hermitian manifold with structure tensors (J, g). The generalized Chern form y is given by

8'ryij = -4JkjP k

- Jkl(v.Jhk)ViJlh-

Now on an almost Kahler manifold

(VkJip)Jj' = (V Jij)Jkp, this is the condition for an almost Hermitian structure to be quasi-Kahler (see e.g., Gray and Hervella [1980]). Using this we have the following computation. 87yijJji = 4T* - Jkl(DiJkh)(ViJhl)Jji

= 4T* - Jkl(ViJkh)(V Jhi)Jli = 4T* - 117JI2,

164

10. Curvature Functionals on Spaces of Associated Metrics

but ,r - r* _ -1 V JI2 and hence 2(r+T*) = 4T* -

jV JI2 = 87ry2;Jiz.

Thus the "total scalar curvature" of an associated metric becomes I(g) _ 47r fm yi, J'i dV. Moreover, writing Q in terms of a J-basis, a simple computation shows that

yiiJ'ZdV = fM

1

2-1 (n - 1)!

JM

yAQ -1.

Thus one has a relatively easy proof that the "total scalar curvature" is a symplectic invariant and writing T - r* as 2T - (r + r*) we see that A(g) and K(g) have the same critical point condition proving Theorem 10.5. On a Kahler manifold the Ricci form is, up to a constant, the first Chern form y, i.e., 27ryij _ -PikJkj. On an almost Kahler manifold the Ricci tensor need not be J-invariant and hence the Ricci form is not in general defined. In [1994] Draghici defined a Ricci form on an almost Kahler manifold by decomposing the Ricci tensor p into its J-invariant and J-anti-invariant parts and defining the Ricci form 0 by z/'(X, Y) = pi''"(X, JY). Similarly define the *-Ricci form by 0*(X,Y) = p*(X, JY); since p*(JX, JY) = p*(Y,X), 0* is a well defined 2-form. Draghici then obtained a cohomological version of the Goldberg conjecture and proved that if M is a compact almost Kahler manifold whose Ricci form is cohomologous to the first Chern class, then M is a Kahler manifold. In the contact case the results corresponding to Theorems 10.4 and 10.5 were obtained by Blair and Ledger in [1986] and the critical point conditions are different. The integral K(g) for a contact manifold M2h+1 is taken to be K(g) = fM r-r*-4n2 dV (see Proposition 7.7) even though it is not necessary to include the constant 4n2. Theorem 10.7 Let M be a compact contact manifold and A the set of metrics associated to the contact form. Then g E A is a critical point of A(g) restricted to A if and only if Q and 0 commute when restricted to the contact subbundle. Theorem 10.8 Let M be a compact contact manifold and A the set of metrics associated to the contact form. Then g E A is a critical point of K(g) if and only if Q - 2nh and 0 commute when restricted to the contact subbundle. Moreover from Proposition 7.7 we see that Sasakian metrics, when they exist, are maxima of K(g). In approaching Theorem 10.6 we remarked that the study of fm T + T* dV in symplectic geometry was motivated by the corresponding study in contact

10.2 The *-scalar curvature

165

geometry (Blair and Perrone [1992]). It is interesting to remark that many results in contact and Sasakian geometry were motivated by the corresponding ones in symplectic and Kahler geometry. Here we have an example of a result in contact preceding its symplectic analogue. In contact geometry the functional 1(g) = fm T + T* dV is not an invariant and gives a critical point problem whose critical point condition gives the important class of K-contact metrics, i.e., associated metrics for which the characteristic vector field generates isometries.

Theorem 10.9 Let M be a compact contact manifold and A the set of metrics

associated to the contact form. Then g E A is a critical point of 1(g) fm T + T* dV if and only if g is K-contact.

Proof. The terms T and T* are differentiated as in Blair and Ledger [1986] and the contributions of p2jDzj cancel instead of doubling up as in Theorem 10.8 and the original proof of Theorem 10.5. The result is that d

dt M

T

T * dV

to

_ -4n

h'1D;l dV. JM

Thus from Lemma 10.2 and h = 0, the critical point condition becomes Oh - hq = 0; but Oh + he = 0 and hence h = 0. Therefore field (see Section 6.2).

is a Killing vector

Turning to the second variation we have the following result of Blair and Perrone [1995].

Theorem 10.10 The index of 1(g) and the index of -1(g) are both positive at each critical point.

Proof. Referring to Perrone and the author [1995] for details, the second derivative of 1(g) evaluated at a critical point is given by the following succinct formula.

I"(0) = 2n

JM

tr(gD£gD) dV.

Now as in the proof of Lemma 10.2, let X1i ... , X2ni l; be a local O-basis defined on a neighborhood U and again note that the first vector field X1 may

be any unit vector field on U orthogonal to l;. Let f be a C°° function with compact support in U and define a path of metrics g(t) as follows. Make no change in g outside U and within U change g only in the planes spanned by Xl and X2 by the matrix

1+t2f2 C

tf

tf1

166

10. Curvature Functionals on Spaces of Associated Metrics

It is easy to check that g(t) E A and clearly the only non-zero components of D are D12 = D21 = f. Denoting the first vector field in the O-basis by X, calculation of 1"(0) in the above formula yields

I"(0) = -4n JM

where X may be regarded as any unit vector field on U belonging to the contact

subbundle. Thus the proof reduces to finding unit vector fields belonging to the contact subbundle on U for which rl([[f, X], X]) has either sign and again we refer to Blair and Perrone [1995] for details. As a application of Theorem 10.10 we prove the following result (Blair and Perrone [1995]).

Theorem 10.11 The functional A(g) restricted to A cannot have a local minimum at any Sasakian metric.

Proof. Suppose that go is a Sasakian metric and a local minimum of A(g) in A. Then there exists a neighborhood U of go E A on which A(go) < A(g). Since all associated metrics have the same volume element, fm To dV < fm T dV for every g E U. From Proposition 7.7,

2T - 4n2 < T + T*

with equality if and only if the metric is Sasakian. Thus we have

I(go) = J 2TO - 4n2 dV < M

f 2T - 4n2 dV < 1(g) M

for every g E U, that is go is a local minimum for I(g) contradicting Theorem 10.10.

10.3

The integral of Ric(e)

The integral, L(g) = fm Ric(e) dV was studied in general dimension by the author in [1984] and independently by Chern and Hamilton in [1985] in the 3-dimensional case. Recall (Corollary 7.1) that

Ric(e) = 2n - trh2; thus K-contact metrics, when they occur, are maxima for L(g) on A. Moreover the critical point question for L(g) is the same as that for fm JhJ2 dV or fm IT12 dV where T (X, Y) _ (££g) (X, Y) = 2g(X, hOY). It is the integral E(g) = fm dV that was studied by Chern and Hamilton [1985] for

10.3 The integral of Ric(C)

167

3-dimensional contact manifolds as a functional on A regarded as the set of CR-structures on M (there was an error in their calculation of the critical point condition as was pointed out by Tanno [1989]). The first result concerning L(g) is the following.

Theorem 10.12 Let M be a compact regular contact manifold and A the set of metrics associated to the contact form. Then g E A is a critical point of L(g) = fm Ric(C) dV if and only if g is K-contact. Proof. As with our other critical point problems, the first step is to compute ae at t = 0 for a path g(t) E A,

dl at

i mk (-h ,,,,h

t=o

-R

i

k TS

r s + 2h ik)Dik dV9.

M

Thus if g(0) is a critical point, Lemma 10.2 gives

Rx

_ -¢2X - h2X + 2hX

as the critical point condition. Using the first formula of Proposition 7.1 this becomes

(Veh)X = -2q5hX. From this we see that the eigenvalues of h are constant along the integral curves of and that for an eigenvalue A 0 and unit eigenvector X, g(VeX, OX) _ - 1. If now M is a regular contact manifold, then M is a principal circle bundle with tangent to the fibres; locally M is U x S'. Since hq + Oh = 0, we may

choose an orthonormal ql-basis of eigenvectors of h at some point of U x S' say, X2i_,, X2i = ¢X2i_1, . Since the eigenvalues are constant along the fibre, we can continue this basis along the fibre with at worst a change of orientation of some of the eigenspaces when we return to the starting point. Thus if Y is a vector field along the fibre, we may write

Y=

c 2i-1X21-1 + 32iX2i) +'y

where the coefficients are periodic functions.

Now suppose that the critical point g is not a K-contact metric. Since 0 and h anti-commute, we may assume that all the A2z_,, i = 1.... , n are non-

negative. Also from (Veh)X = -20hX it is easy to see that if some of the A2i_, vanish, the zero eigenspace of h is parallel along and hence we may choose the corresponding X2i_, and X2i parallel along a fibre. Again since M is regular we may choose a vector field Y on U x Sl such that at least some

168

a2i-1

10. Curvature Functionals on Spaces of Associated Metrics

0 for some A2i_1 ; 0 and Y is horizontal and projectable, that is

Y] = 0. Writing Y = >i(a2i-iX2i-i + 132iX2i) along a fibre we have, using

tt

((Sa2i-1)X21-1 + a2i-1V6X2i-1 + (732i)X2i + 32iVi;X2i

+a2i-1X21 + A2i-1a2i-1X21 - N2iX2i-1 + AN-AiX2i-1)

Taking components we have 0 = Sa2j-1 +

0 = 732j +

a2i-1g(VSX2i, X2j) + )'2j-1132j,

/32ig(VEX2i,X2j) +''2j-la2j-1

Multiplying the first of these by /32ji the second by a2j-1 and summing on j we have 2

2j-102j )

=-

2

i-1(a2j-1 + P2j)

0.

Thus Ej a2j_1f2j is a non-increasing, non-constant function along the integral curve, contradicting its periodicity.

One might conjecture Theorem 10.12 without the regularity, however we have the following counterexample: The standard contact metric structure on the tangent sphere bundle of a compact contact manifold of constant curvature

-1 is a critical point of L but is not K-contact (the author [1991b]). We give this in the next theorem. Recall also the result of Tashiro, Theorem 9.3,

that the standard contact metric structure on the tangent sphere bundle of a Riemannian manifold is K-contact if and only if the base manifold is of constant curvature +1. Theorem 10.13 Let T1M be the tangent sphere bundle of a compact Riemannian manifold (M, G) and A the set of all Riemannian metrics associated to its standard contact structure. Then the standard associated metric is a critical point of the functional L(g) if and only if (M, G) is of constant curvature

+1 or -1. Proof. We have seen that the critical point condition of L(g) is

Rx £ _ -q2X - h2X + 2hX or 02 + h2 + l = 2h. Applying this to a vertical vector U E TtT1M as in the proof of Lemma 9.2 we find that (M, G) is locally symmetric by the Lemma

10.3 The integral of Ric(h)

169

of Cartan [1983, pp. 257-258] and that LtX = X for any orthonormal pair {X, t} on (M, G). Thus the only eigenvalues of Lt [t]1 --+ [t]1 are ±1. Now M is irreducible; for if M had a locally Riemannian product structure, then choosing t tangent to one factor and X tangent to the other, we would have Rx tt = 0, contradicting the fact that the only eigenvalues of Lt are f 1. :

Now the sectional curvature of an irreducible locally symmetric space does not change sign. Thus if for some t, Lt had both +1 and -1 as eigenvalues, there

would be sectional curvatures equal to +1 and -1. Consequently only one eigenvalue can occur and hence (M, G) must be a space of constant curvature

+1 or -1. Conversely if (M, G) has constant curvature c, our expressions for h on horizontal vectors X orthogonal to and vertical vectors U, namely

hXt = -Xt + (Rxtt)H, hUt = Ut - (Rxvtt)V become hX = (c - 1)X and hU = (1 - c)U. Similarly lU = c2U and 1X = (4c - 3c2)X. Substituting these into the critical point condition we see that it is satisfied if and only if c = +1.

In [1991] S. Deng studied the second variation of the functional L(g) or equivalently, of E(g) = fm IT12 dV.

Theorem 10.14 Let g E A be a critical point of E(g), then g is a minimum.

Proof. As with Theorem 10.10 we will sketch the proof and refer to Deng [1991] for details. The first step is a lengthy calculation of E"(0) yielding

E"(0) = 2 f I££D12 dV > 0. M

The second step is an auxiliary result that if LCD = 0, then ITI2 is constant along the geodesics g(t) = geDt in A. Now proceed as follows. If 0 for all D, then ITS is constant and hence E(g) is constant. So suppose that g is not a minimum and that there exist D such that £CD 4 0. Let V be the subspace of T9A consisting of those D with ££D = 0 and V1 its orthogonal complement. Let U be a neighborhood of g in exp9V1 (cf. Ebin [1970, p. 37]) on which E exceeds E(g). Let W be a neighborhood of U C A formed by geodesic arcs in directions belonging to V. Since g is not a minimum, there exists g E W such that E(g) < E(g). Then the geodesic y from g in direction D E V meets U at say g and E is constant along y. Therefore E(g) < E(g) contradicting E(g) > E(g) for g E U.

Little has been said about the existence of critical metrics and in general this is a difficult problem. Recall the notion of an almost regular contact manifold as given by C. B. Thomas [1976] (Section 3.4). Rukimbira [1995c]

10. Curvature Functionals on Spaces of Associated Metrics

170

proved that every almost regular contact manifold admits a critical metric for E(g) (equivalently L(g)). Finally in [1992b] D. PerroneIM considered the integral

F(g) =

+ T* + 2nRic(e) dV

F(g) = fm 7- + ri dV on A. In dimension 3, letting K(D) denote the sectional curvature of the contact subbundle D, and r* = 2K(D), so Tl =,r. In higher dimensions using T = 2K(D) +

or setting Ti = T* + Proposition 7.7,

2n) +

Ti = T + (2n -

2

(JVOI'

- 4n)

and hence if the manifold is Sasakian, Ti = T. The critical point condition for

F(g) is

10.4

0.

The Webster scalar curvature

In Theorem 6.6 we saw that a contact metric manifold is a strongly pseudoconvex CR-manifold if and only if (Vxq1)Y = g(X+hX,Y)1;-r7(Y)(X+hX). On a strongly pseudo-convex CR-manifold Tanaka [1976] introduced a canonical connection. In [1989] Tanno introduced the corresponding connection on a contact metric manifold called the generalized Tanaka connection; it agrees with the connection of Tanaka when the contact metric manifold is a strongly pseudo-convex (integrable) CR-manifold. This connection, *V, is defined by

*VXY = VxY + i7(X)OY - i (Y)Vx + (Vxii)(Y)e = VxY + rl(X )qY + i7(Y) (qlX + chX) + dri(X, Y)C + drl(hX,

The torsion of this connection *T is then

*T(X,Y) = rl(X)gY - i7(Y)OX - ri(Y)Vx + i7(X)VyE + 2drl(X,Y) = r(Y)OhX - r7(X )qhY + 2g(X, qY)e. Tanno [1989] then proves the following proposition.

Proposition 10.1 The generalized Tanaka connection *V on a contact metric manifold is the unique linear connection such that

*Vg=0

*Vr7 =0,

(*VxO)Y = (VxO)Y - g(X + hX, Y) + rl(Y)(X + hX), *T

OY) = -O*T

*T(X, Y) = 2di7(X, Y)1;,

Y), on D.

10.4 The Webster scalar curvature

171

Tanno also computes the curvature of *V and upon contraction obtains the generalized Tanaka-Webster scalar curvature

Wl = T - Ric(h) + 4n. This is 8 times the Webster scalar curvature as defined by Chern and Hamilton [1985] on 3-dimensional contact manifolds and as used by Perrone [1998] in Theorem 7.21. We now prove a theorem of Chern and Hamilton [1985], an alternate proof of which was given by Perrone in [1990], and a theorem of Tanno [1989]. The proofs will be given simultaneously.

Theorem 10.15 (Chern-Hamilton) Let M be a compact 3-dimensional contact manifold and A the set of metrics associated to the contact form. Then g E A is a critical point of El (g) = fm Wl dV if and only if g is K-contact.

Theorem 10.16 (Tanno) Let M be a compact contact manifold and A the set of metrics associated to the contact form. Then g E A is a critical point of El (g) = fm Wl dV if and only if

oQql)06. Proofs. Clearly it is enough to consider fm T - Ric(1;) dV and having computed the derivatives of each term separately in our previous theorems we have

dt f T - Ric(h) dV M

(_Pki + hir,,,h""'k + = f M t=0

-2h ik )Dik dV.

Thus by Lemma 10.2 we see that the critical point condition is

(Qc - OQ) - (l0-0l)

(77oQO) ®6.

Now in dimension 3, the Ricci operator determines the full curvature tensor, i.e.,

RxyZ = (g(Y, Z)QX - g(X, Z)QY + g(QY, Z)X - g(QX, Z)Y) -2 (g(Y, Z)X - g(X, Z)Y). Therefore the operator l is given by

IX = QX - 77(X )W +

)X - g(QX, ) - 2 (x - rl (X

from which

(l0 - Ol)X = (QO - cQ)X + i(X)cQ - g(Qq X, 0-

172

10. Curvature Functionals on Spaces of Associated Metrics

Combining this and the critical point condition we have 4qh = 0 and hence, since he = 0, h = 0.

As we saw in the closing remarks of the last section, in dimension 3 T and T* = 2K(D). Using this, W1 = T - Ric(Z;) H- 4n becomes 2K(D) W1 =

2(T+T*+8).

Thus in dimension 3 the critical point problem for E1(g) is the same as the critical point problem for 1(g) in Theorem 10.9, suggesting that

W = 2(r+7-*+4n(n+l)) may be the proper generalization of the Webster scalar curvature. There are however other generalizations of the Webster scalar curvature. Again in dimension 3, we may write W1 as T* + Ric(h) + 4 and we define a second generalization of the Webster scalar curvature by W2 = T* + Ric(t;) + 4n2.

In general dimension the critical point condition of E2(g) = fm W2 dV is (Qq5-OQ)- (l0-01) = -4(2n - 1)0h - 77

(see Blair and Perrone [1992]). The generalization of the Webster scalar curvature W = 2 (r+T*+4n(n+ l)) is the average of W1 and W2. Th. Koufogiorgos [1997b] has considered the difference of these as we briefly mention his results.

Proposition 10.2 Let Men+1 be a contact metric manifold with a strongly pseudo-convex CR-structure. Then

Theorem 10.17 Let Men+1 be a contact manifold with n > 1 and A the set of associated metrics. If g E A gives rise to a strongly pseudo-convex CRstructure, then g is a critical point of fmW, - W2 dV. The idea of the proof of this theorem is to find the critical point condition of fm W1 - W2 dV which is exactly the formula of Proposition 10.2. Finally a word is in order on the constants depending on dimension which occur in the definitions of W1, W2 and W. Again recall the notion of a Dhomothetic deformation, namely a change of structure tensors of the form 1

=a77,

=1

,

_O, 9=ag+a(a-1)77®77

10.5 A gauge invariant

173

where a is a positive constant. By direct computation one shows that T, Ric(h) and T* transform in the following manner. 2

T= r+ Ric(h) =

a2

(Ric(h) + 2n(a2 - 1))

a + 1 a2a a From these we seea that W1 = a W1 i W2 = a W2 and 4V = W.

r* + a a2 1

T* =

2n (2n (1

10.5 A gauge invariant Use has often been made of the notion of a D-homothetic deformation as we have seen. The notion of a gauge transformation of a contact metric structure has not received as much attention. No doubt this is due in part to the computational complexity; nonetheless, comparing with the notion of a contact structure in the wider sense, the notion of a gauge transformation should be fundamental and deserving of attention. Let (M2-+1, 0, , ,q, g) be a contact metric manifold and consider a gauge transformation of the contact structure, g = Qi' where v is a positive function on Men+1. Let (='OVo-, Vo being the g-gradient of the function o and let z be the covariant form of Now define new structure tensors e, and g by CT

0-(g-,zz(g

2o-

®r7.

is a contact metric structure and the change from (0, Then (0, g) to (0, , r7, g) is called a gauge transformation of the contact metric structure. When v is a constant this is a D-homothetic deformation. The notion of a gauge transformation in contact metric geometry is due to Sasaki [1965, p. PTSB-1-3] (see also Tanno [1989]). Tanno [1989] computed the change of the generalized Tanaka-Webster scalar

curvature under a gauge transformation. The result of his computation is (W1 2(na 1) (n + 1)2n - 2) W1 = 0-

-

(AO' - O') _

(I da12 -

In the same paper Tanno generalizes an invariant of Jerison and Lee [1984] for strongly pseudo-convex CR-manifolds; specifically he considers

f

K(o,g) = inf

f

M

(

4(n + 1) n (I df I2

- (6f )2) + W1 f 2) dVg

174

10. Curvature Functionals on Spaces of Associated Metrics

where the infimum is taken over all non-negative functions f such that fm f P dV9 =

1, p = 2 + 2 and proves that r,l,r,gl is a gauge invariant. Now for a compact contact manifold (M2n+1, i) let .T(p) be the set of all non-negative functions f such that fm f P dVg = 1 and define a functional F,, : M x F(p) ---> R by

,,g, f) = fM \4(nn l) (ldf I2 - (e.f)2) + Wl f2) dVg. Also define F(,r,g) : 1 (p) --* R by F(,,,) (f) = Fr (g, f ). In [1989] Tanno studied

these functionals in detail; here we mention only the following result.

Theorem 10.18 Let (M2n+1, 0, p g) be a compact contact metric manifold with constant generalized Tanaka-Webster scalar curvature W1. Then F. is critical at the pair (g, f) with f = (vol(M, g))-1/P, if and only if (Qb-OQ)-(l0-¢l)

Note that this is also the critical point condition of E1 (g) = fm W1 dV in Theorem 10.16.

10.6

The Abbena metric as a critical point

In Section 1.1 we discussed Thurston's example of a compact symplectic manifold with no Kahler structure and a natural Riemannian metric on this manifold introduced by E. Abbena [1984]. She computed the curvature and with respect to the basis {e2} introduced in Section 1.1 and the Ricci operator Q is given by the matrix 0

0

10 0

0

-2 0 0 0

2 0

0

0 0 0

From the expression for Q it is clear that (M, g) is not Einstein nor is QJ = JQ. Thus this metric is not a critical point for A(g) considered as a functional on M1 or on A or for K(g) on A. Also r = -1/2 and r* = +1/2 giving zero for the "total scalar curvature". In [1996] Park and Oh discussed a functional for which the Abbena metric on the Thurston manifold is a critical point; their results are given in the following theorem. Recall that M1 denotes the space of Riemannian metrics with unit volume.

Theorem 10.19 The Abbena metric on the Thurston manifold is a critical point of the functional

Ivt (3 trQ3 -

) dVg

10.6 The Abbena metric as a critical point

175

on MI. The index of this functional and its negative are both positive at the Abbena metric on the Thurston manifold. We remark that the Abbena metric on the Thurston manifold is a critical point for K(g) in a different context. C. M. Wood [1995] showed that the Abbena metric is a critical point of K(g) = -2 fM defined with respect to variations through almost complex structures J which preserve g. For this problem the critical point condition is [J, V*VJ] = 0,

where V*VJ is the rough Laplacian of the metric in question.

11 Negative e-sectional Curvature

11.1

Special Directions in the contact subbundle

The purpose of this chapter is to introduce some special directions that belong to the contact subbundle of a contact metric manifold with negative sectional curvature for plane sections containing the characteristic vector field or more generally when the operator h admits an eigenvalue greater than 1; these directions were introduced by the author in [1998]. We also discuss in this chapter some questions concerning Anosov and conformally Anosov flows. For simplicity we will often refer to the sectional curvature of plane sections containing the characteristic vector field as e-sectional curvature. We may regard the equation

Vx = -OX - OhX of Lemma 6.2 as indicating how

or, by orthogonality, the contact subbundle,

rotates as one moves around on the manifold. For example when h = 0, as we move in a direction X orthogonal to , is always "turning" or "falling" toward -OX. If hX = AX, then Vxe = -(1-I- A)qX and again l;' is turning toward -OX if A > -1 or toward OX if A < -1. Recall that we noted above that if A is an eigenvalue of h with eigenvector X, then -A is also an eigenvalue with eigenvector OX.

Now one can ask if there can ever be directions, say Y orthogonal to along which l;' "falls" forward or backward in the direction of Y itself.

Theorem 11.1 Let Men+1 be a contact metric manifold. If the tensor field

h admits an eigenvalue A > 1 at a point P, then there exists a vector Y

178

11. Negative -sectional Curvature

orthogonal to

at P such that Vy is collinear with Y. In particular if M2n+1

has negative f-sectional curvature such directions Y exist.

Proof. Let \ denote a positive eigenvalue of h and X a corresponding unit eigenvector. Then

Vxl: = -(1 + A) OX,

V ox = (1 - A)X.

Now let Y = aX + bOX with a > 0, b > 0, a2 + b2 = 1 and suppose that

Vy = aY. Then

a(aX+bqX)=Vy =-(1+.\)agX+(1-A)bX from which as = (1 - A)b, ab = -(1 + A)a and hence a

2

A- 1

2

A+ 1

b

2A

2A

a=-A

2

1.

Thus we see that directions along which Vy is collinear with Y exist whenever

h admits an eigenvalue greater than 1. From the fact that Ric(l;) = 2n - trh2 (Corollary 7.1) we see that if M27,+1 has negative c-sectional curvature, at least one of the eigenvalues of h must exceed 1. Note that when there exists a direction Y along which Vye is collinear with Y as above, there is also a second such direction, namely Z = aX - bOX. For Z we have Vze = -aZ; thus we think of as falling backward as we move in the direction Y and falling forward as we move in the direction Z. Next note that g(Y,Z)=a2-b2=-1

A

and hence that such directions Y and Z are never orthogonal. Also if A has multiplicity m > 1, then there are m-dimensional subbundles Y and Z such

that Vy = aY for any Y E Y and Vze = -aZ for any Z E Z. We refer to directions along which the covariant derivative of t; is collinear with the direction as special directions.

11.2

Anosov flows

The most notable example of a contact manifold for which the characteristic vector field is Anosov is the tangent sphere bundle of a negatively curved manifold; here the characteristic vector field is (twice) the geodesic flow as we saw in Section 9.2. In the case of the tangent sphere bundle of a surface, this is closely related to the structure on SL(2, R) from both the topological and Anosov points of view. If we set Z2 = { (o o) ( 1 0) }, then PSL(2, R) _

11.2 Anosov flows

179

SL(2, R)/Z2 is homeomorphic to the tangent sphere bundle of the hyperbolic plane. Moreover the geodesic flow on a compact surface of constant negative curvature maybe realized on PSL(2, R)/F by { (o e°_t) }, where F is a discrete subgroup of SL(2, ][8) for which SL(2, R)/I' is compact, (see e.g., Auslander, Green and Hahn [1963, pp. 26-27]). However from the Riemannian point of view these examples are quite different as we shall see. In fact in the case of the tangent sphere bundle of a negatively curved surface, the special directions never agree with the Anosov directions (Theorem 11.3). Classically an Anosov flow is defined as follows (Anosov [1967]). Let M be a compact differentiable manifold, a non-vanishing vector field and {zit} its 1parameter group of (Ck) diffeomorphisms. {Ot} is said to be an Anosov flow (or to be Anosov) if there exist subbundles Es and Eu which are invariant along

the flow and such that TM = ES ® Eu ® { } and there exists a Riemannian metric such that lzbt*Yl < ae-"JYJ fort > 0 and Y E Ep,

IV)t*Yl 1. Further by way of notation, set

xv-yu=1).

SL(2, ll8) =

Now consider the matrices

a2-1

2

0

-2 a2-1 ),

0 0

-2 +1

a21

-2 a-1

0 '

0

-

a21

0

in the Lie algebra .s((2, R) which we regard as the tangent space of SL(2, R)

at the identity. Applying the differential of left translation by

x C

u vy

to

these matrices gives the vector fields

Si=2 A2-1 Ix- -y y+ua

-v0 a al yax-x aa +vau-uav)

r+1 a S2=

2

s=- a-1( a 2

y

al

a a yax+xa+vau+uav y

whose Lie brackets satisfy (*). Using these matrices again, define a left invari-

ant metric g; then {(i, S2, 31 is an orthonormal basis. The contact form wl we denote by y and it is given by 77 _

2 2

- 1(vdx - ydu).

182

11. Negative C-sectional Curvature

The characteristic vector field C is S1. g is an associated metric and 0 as a skew-symmetric operator is given by ¢ = 0 and ¢S2 = S3. The symmetric operator h is given by hC = 0, h(2 = a(2i h(3 = -.X53. The special directions are

Y and

__

V

a+1

a-1r

S2+

2A

2.,

-1rS2 rA Z=

2a

(s-S3 =

a2-1

a

al

xay+uav J

V

C

a2 -1

a 8 Cyax + vau)

The 1-parameter group of {fit} of C is given by

xe2 a -1 t ye-2 a -1 t

xy U

v)

=

ue2 a -1t ve-2 a -1t

Then

'fit*Y = -

2_

le-2

/\2

t*Z =

a -1t

1 a -1t

e2

Cx y

+u-) _ (e-2

8 + va ( y ax 8u)

=

t)Y

(el t) Z.

Thus the corresponding subbundles 32 and .Z are invariant under the flow. + a±1 = 1 and hence Finally since {(i, S2, (3} is orthonormal, y12 = a-1 2a 2a

Ot*YI=(e-2

V'1'_-1t )1Y1;

similarly

IIt*ZI = (e2 a -1t) ZI. Thus C is an Anosov vector field and the special directions Y and Z agree with the Anosov directions.

In contrast to this, consider the contact metric structure on the tangent sphere bundle. We mentioned at the beginning of this section that the tangent sphere bundle of a surface is closely related to the structure on SL(2, IR) from both the topological and Anosov points of view. Comparing the above with the following theorem we see that from the Riemannian point of view these are quite different.

Theorem 11.3 With respect to the standard contact metric structure on the tangent sphere bundle of a negatively curved surface, the characteristic vector field is Anosov, but the special directions never agree with the stable and unstable directions.

11.2 Anosov flows

183

Proof. We noted in Section 9.2 that for the standard contact metric structure on the tangent sphere bundle of a Riemannian manifold, the characteristic vector field is (twice) the geodesic flow, which is an Anosov vector field when the base manifold is negatively curved (see e.g., Anosov [1967]). By Theorem

11.2 if the special directions of the contact metric structure agree with the Anosov directions, then Veh = 0. Now by Theorem 9.5 the standard contact metric structure of the tangent sphere bundle of any Riemannian manifold satisfies 0£h = 0 if and only if the base manifold is of constant curvature 0 or +1. Perrone [2000], utilizing the special directions Y and Z belonging to the contact subbundle on a contact metric manifold of negative e-sectional curvature, introduced another notion. Let Y and Z denote the subbundles generated by Y and Z. The special directions are said to be Anosov-like if the subbundles and Z ® {l;} are integrable. Perrone then proved that a contact mety® ric 3-manifold admits Anosov-like special directions if and only if it satisfies

Ooh = 0 and has negative Ricci curvature in the direction . Three dimensional contact metric manifolds satisfying this condition were called 3-7--manifolds by F. Gouli-Andreou and Ph. Xenos [1998]. The name comes from the equivalent condition Q£T = 0 where here T = £Cg; in particular T and h are related by T(X, Y) = 2g(h¢X, Y). A 3-dimensional contact metric manifold on which the Ricci operator Q and 0 commute satisfies Veh = 0 but not conversely. We close this section with an example (the author [1996]) of a contact metric manifold satisfying 0£h = 0 but with Qq # qQ. We include this example here as it is also an example of a metric on R3 with respect to which the coordinate field aZ is Anosov.

Consider the standard Darboux contact form ri = 2 (dz - ydx) and characteristic vector field = 2-L. . Let f be a smooth function of x and y bounded below by a positive constant c. Then the metric given by

g_

1

4

e=f+(1+f2)e-=f-2 +y2

e=f 1

-y

esf-1

e zf

0

-y

0

1

f

is an associated metric. The tensor fields 0 and h are given by esf-1

-(exf+(1+f2)e sf-2) y(ex

f

1)

f

yezf

184

11. Negative c-sectional Curvature eZf

h

(fe=f+(1+f2)(-f)e-If)

=

yezf

f ezf

0

-ezf

0

yf eZf

0

By direct computation V h = 0. Also 2A2 = trh2 = 2(1 + f2) and hence the positive eigenfunction of h is A = 1 + f 2 > 1. Now on a 3-dimensional contact metric manifold satisfying Q¢ = OQ, the eigenfunction A is a constant (Blair, Koufogiorgos, and Sharma [1990]). Thus if f is not constant this structure on R3 satisfies 0£h = 0 but not QO = qQ. For this structure the special directions discussed in Section 11.1 are given by

Y=fix- +yfa

a

(9

,

Z=ay.

To check that = 2a is Anosov with respect to g, consider for simplicity just aZ; its flow 'fit maps a point Po(x, y, z) to the point P(x, y, z + t). Now recalling that the function f was chosen to be bounded below by a positive constant c, we have for t < 0,

a ilft*

ay

'(P) = 2 e

(Po)

(s+t )f 2

=

tf

e2

ay

(P°) < e z ay (P°)

.

Similarly for t > 0, t*

a_a

(f ax

al

ay + yf az (P0)

e

r

a_a

\f ax

al

ay + yf 8z

(PD)

Thus aZ, equivalently l;, is Anosov with respect to this metric; Y determines the stable subbundle and Z the unstable subbundle.

11.3

Conformally Anosov flows

Recently there has been interest in a notion more general than an Anosov flow, namely, a conformally Anosov flow. We will show that if a compact contact metric 3-manifold has negative c-sectional curvature, then the characteristic vector field is conformally Anosov. Mitsumatsu [1995] and Eliashberg and Thurston [1998] introduced a generalization of Anosov flows as follows. A flow fit or its corresponding vector field is said to be conformally Anosov, Eliashberg and Thurston [1998] (projectively Anosov Mitsumatsu [1995]) if there is a continuous Riemannian metric and a

continuous, invariant splitting, TM = ES ® E' ® {6} as in the Anosov case such that for Z E E" and Y E E3, IV)t* Z I > eCt

I?

Ot*YI

IYI

11.3 Conformally Anosov flows

185

for some constant c > 0 and all t > 0. Now a contact structure rl on a 3-dimensional contact manifold M3 determines an orientation on M3. This is true in dimension 3 even for a contact structure in the wider sense, since the sign of rl A dry is independent of the choice of local contact form 17. The main result for our purpose from Mitsumatsu [1995, p. 1418] and Eliash-

berg and Thurston [1998, pp. 26-27] is the following.

Theorem 11.4 If two contact structures (in the wider sense) on a compact 3-dimensional contact manifold M3 induce opposition orientations, then the vector field directing the intersection of the two contact subbundles is a conformally Anosov flow. Conversely given a conformally Anosov flow on M3, there exist two contact structures giving opposite orientations on M3 whose contact subbundles intersect tangent to the flow.

We now show that certain curvature hypotheses on a compact contact metric 3-manifold imply that the characteristic vector field l; is conformally Anosov; in particular negative c-sectional curvature is such a hypothesis (the author [2000]). A variation of this result appears in Blair and Perrone [1998].

Theorem 11.5 Let (M3, 0, C,17, g) be a compact 3-dimensional contact metric manifold with nowhere vanishing h and {e1, e2(= qel), } an orthonormal eigenvector basis of h with he1 = Xe1 and A the positive eigenvalue. If e1) < (1 +A)2 and K(1;, e2) < (1 - A)2, then 6 is conformally Anosov. In particular, if the e-sectional curvature is negative, t; is conformally Anosov.

Proof. Let w1, w2 be the dual 1-forms of e1 and e2. Since the three eigenvalues of h are everywhere distinct, the corresponding line fields are global. By the orientability, one may choose local bases directing the line fields that either agree or have two directions reversed in the overlap of coordinate neighborhoods. Thus in computing w1 A dw1 and w2 A dw2 on such a basis we may regard the computations as global. By straightforward computation we have w1 A dw1(el, e2,

)=

w2 Adw2(e1i e2, C) _

A-1

-

-

6

6

1

1

e2),

- 6g(VEe1 e2)

On the other hand, applying the first equation of Proposition 7.1 to e1 and e2 we obtain e1) = 1 - A2 - 2Ag(Vee1, e2), e2) = 1 - A2 + 2Ag(VCe1, e2).

Combining these equations we have w1 A dw1(e1i e2,

)=

12A ((A

-

1)2

- K(e, e2))

186

11. Negative k-sectional Curvature

and

w2 A dw2(ei, e2, )

=

12A

(-(A -I- 1)2

+ K( el))

The hypotheses now imply that wl A dw'(el, e2i ) > 0 and w2 A dw2(el, e2, ) < 0. Therefore is conformally Anosov by the result of Mitsumatsu and Eliashberg-Thurston.

We remark that the curvature of the standard contact metric structure on the tangent sphere bundle of a surface of negative curvature satisfies the curvature hypotheses in the first statement of the theorem. In particular the standard contact metric structure on the tangent sphere bundle of a surface of constant curvature -1 has sectional curvature -7 for horizontal plane sections and sectional curvature +1 for plane sections spanned by 6 and the vertical direction; the positive eigenvalue A is +2 with vertical vectors as eigenvectors (see Section 9.2). We also remark that our argument uses two contact structures to study a third, an interesting idea in view of the result of Gonzalo [1987] (Section 3.2) that a 3-dimensional compact orientable manifold admits three independent contact structures. It is immediate that the characteristic vector field of a contact structure can never be Anosov or conformally Anosov with respect to a Sasakian metric. This is a consequence of the fact that on a Sasakian manifold 6 is a Killing vector field; thus its flow is metric preserving and therefore cannot satisfy the exponential growth behavior in the definition of a classical or conformal Anosov flow. It is possible however for a vector field to belong to the contact subbundle of a Sasakian structure and be conformally Anosov with respect to this metric. We present an example of this which has the additional feature of the example that the characteristic vector field is invariant along the flow (again see Blair and Perrone [1998]). We begin with the following lemma. Lemma 11.1 On a 3-dimensional contact manifold in terms of local Darboux coordinates (x, y, z) (rl = l (dz - ydx)) any associated metric is of the form

g4 1

a

b

-y

b

c

0

-y 0

1

with ac - b2 - cy2 = 1; the metric is Sasakian if and only if the functions a, b and c are independent of z.

Proof. The form of the last row and column follow from the requirement that rl(X) = In dimension 3 the remaining requirements reduce to the determinant of the matrix (without the 1) being 1. Also in dimension

11.3 Conformally Anosov flows

187

3 the Sasakian condition is equivalent to the contact metric structure being K-contact. Thus evaluating the Lie derivative, ££g, on the coordinate vector fields we see that = 2a is Killing if and only if the functions a, b and c are independent of z.

(x, y, z) l y > 0} with the stanTo construct the example, consider R+ dard Darboux contact form y = (dz - ydx). The characteristic vector field is a 2L and the Riemannian metric given by the following matrix is Sasakian by Lemma 11.1 e2y

Ve-2y

g= 4- I /e2y-y2 -y

- y2 - 1 -y 1

0

0

1

The vector field -iL is conformally Anosov with respect to this metric. To see this we observe that the subbundles determined by a = a and aL correspond to E9 and E" respectively. The flow simply maps a point Po(x, y, z) to the point P(x, y + t, z) and we have easily that l a (P) I l a (P) I

_ et

ax (Po)

I a (Po)I

Clearly this flow satisfies Ot,, = . For a discussion of conformally Anosov flows on 3-dimensional homogeneous

contact metric manifolds including an Anosov flow belonging to the contact subbundle on the Lie group E(1, 1), see Blair and Perrone [1998].

12 Complex Contact Manifolds

12.1

Complex contact manifolds and associated metrics

While the study of complex contact manifolds is almost as old as the modern theory of real contact manifolds, the subject has received much less attention and as many examples are now appearing in the literature, especially twistor spaces over quaternionic Kahler manifolds (e.g., LeBrun [1991], [1995], Moroianu and Semmelmann [1994], Salamon [1982], Ye [1994] ), the time is ripe for another look at the subject. As an indication of this interest we note, for example, the following result of Moroianu and Semmelmann [1994] that on a compact spin Kdhler manifold M of positive scalar curvature and complex dimension 41 + 3, the following are equivalent: (i) M is a Kahler-Einstein manifold admitting a complex contact structure, (ii) M is the twistor space of a quaternionic Kahler manifold of positive scalar curvature, (iii) M admits Kahlerian Killing spinors. LeBrun [1995] proves that a complex contact manifold of positive first Chern class, i.e., a Fano contact manifold, is a twistor space if and only if it admits a Kahler-Einstein metric and conjectures that every Fano contact manifold is a twistor space. Specifically the notion of a complex contact manifold stems from the late 1950s and early 1960s with the papers of Kobayashi [1959] and Boothby [1961],

[1962]; this is just shortly after the Boothby-Wang fibration in real contact geometry. Then in [1965], J. A. Wolf studied homogeneous complex contact manifolds and their relation to quaternionic symmetric spaces. In the 1970s and early 1980s there was a development of the Riemannian theory of complex contact manifolds by Ishihara and Konishi [1979], [1980], [1982]. In this development however, the notion of normality seems too strong since it precludes

190

12. Complex Contact Manifolds

the complex Heisenberg group as one of the canonical examples, although it does include complex projective spaces of odd complex dimension as one would

expect. In the real case both the Heisenberg group and the odd-dimensional spheres have natural Sasakian (normal contact metric) structures. As a subject the Riemannian geometry of complex contact manifolds is just in its infancy. For brevity, in this chapter we will give more of a survey of results and refer to the references for proofs. A complex contact manifold is a complex manifold of odd complex dimension 2n+1 together with an open covering {0a} by coordinate neighborhoods such

that: 1. On each Oa there is a holomorphic 1-form B« such that Ba A (do,, )n 54 0;

2. On Q. n Op # 0 there is a non-vanishing holomorphic function fap such

that 0. = fap0p.

The subspaces {X e TO,,, B« (X) = 0} define a non-integrable holomorphic subbundle 7-l of complex dimension 2n called the complex contact subbundle or horizontal subbundle. The quotient L = TM/7-l is a complex :

line bundle over M. Kobayashi [1959] proved that cl (M) = (n + 1)cl (L) and hence for a compact complex contact manifold, a complex contact structure is given by a global 1-form if and only if its first Chern class vanishes (see also Boothby [1961], [1962]). It is for this reason that our definition of complex contact structure is analogous to that of a contact structure in the wider sense. Even for the most canonical example of a complex contact manifold, CP2n+1, the structure is not given by a global form. Since a holomorphic p-form on a compact Kahler manifold is closed (see e.g., Goldberg [1962, p. 177]), no compact Kahler manifold has a complex contact structure given by a global contact form. Moreover, Ye [1994] showed that a compact Kahler manifold with vanishing first Chern class has no complex contact structure. There are however interesting examples of complex contact manifolds with global complex contact forms as we shall see; these are called strict complex contact manifolds (Foreman [2000a], [2000b]). We will not need a complex Darboux theorem in our development here, but a complex version of the real Darboux theorem is possible and such a result is discussed briefly by LeBrun [1995, p. 423]. If (M, {O }) is a complex contact manifold, the transition functions fap

define a holomorphic line bundle over M, viz., L'. Using local sections of this bundle we define complex-valued 1-forms {ira} such that each Ira is a non-vanishing, complex-valued function multiple of Ba and on 0a n 0p # 0,

7a =

hap : Oc. n Op

S1.

12.1 Complex contact manifolds and associated metrics

191

The hap are then the transition functions of a circle bundle P over M (see Ishihara and Konishi [1982]) and on Oa, Ira n (d7ra)' n Ira n (dTra)T

0.

Writing Ira = ua - iva, va = ua o J since wa is holomorphic. Moreover ua and va transform naturally with respect to S1, namely if hap = a + ib,

up = aua - bva,

vp = bua + ava,

a2 + b2 = 1.

The set {Ira} is called a normalized contact structure with respect to {Ba}, Foreman [1996].

For simplicity we will often omit the subscripts on the local tensor fields. Define a local section U of TM, i.e., a section of TO, by du(U,X) = 0 for every X E 7-l, u(U) = 1 and v(U) = 0. Such local sections then define a global

subbundle V by Vlo = Span{U, JU}. We now have TM = 7-l ® V and we denote the projection map to 7-l by p : TM -----> H.

We assume throughout that V is integrable and we call V the vertical subbundle or characteristic sub bundle. On the other hand if M is a complex manifold with almost complex structure J, Hermitian metric g and open covering by coordinate neighborhoods {Oa}, M is called a complex almost contact metric manifold if it satisfies the following two conditions:

1) On each Oa there exist 1-forms ua and va = ua o J with orthogonal dual vector fields Ua and Va = -JUa and (1,1) tensor fields Ga and Ha = GaJ such that

G201 =H2a=-I+ua®Ua+va®Va, GaJ = -JGa, 2) On Oa n Op

GaU = 0,

g(X, GaY) = -g(GcX, Y),

aua - bva,

vp = bua + ava,

0,

up =

Gp = a Ga - bHa,

Hp = bGa + aHa

where a and b are functions with a2 + b2 = 1. Returning to the local forms Ira = ua - iva on a complex contact manifold,

a Hermitian metric g is called an associated metric if there are local fields of endomorphism Ga such that the tensor fields Ga, Ha = G« J, Ua, Va = -JUa, ua, va, g form a complex almost contact metric structure satisfying g(X, GaY) = dua (X, Y),

g(X, HcY) = dva(X, Y),

X, Y E 7i .

192

12. Complex Contact Manifolds

As a consequence we also have Up = aUc. - bVa, VQ = bUa + aVa and hence Ua + iV0, = h-p (U,3 + iVR). In particular { h-1 } are the transition functions of the bundle V. Kobayashi [1959] also showed that the structural group of a complex contact

manifold is reducible to (Sp(n) U(1)) x U(1). Such a reduction is equivalent to a complex almost contact metric structure, cf. Shibuya [1978]. Shibuya's approach is the complex analogue of that of Hatakeyama [1962] which we utilized in Section 4.2. A formulation in terms of global tensor fields similar to the fundamental 4-form of a quaternionic Kdhler manifold (see e.g., Ishihara [1974]) was given by Blair, Ishihara, and Ludden [1978]. Ishihara and Konishi [1982] (see also Foreman [1996]) prove that a complex contact manifold admits a complex almost contact metric structure for which the local contact form 0 is of the form u-iv to within a non-vanishing complexvalued function multiple and the local tensor fields G and H are related to du and dv by

du(X, Y) = g(X, GY)+(onv)(X,Y), dv(X, Y) = g(X, HY)-(o Au)(X,Y) where or(X) = g(VxU, V). We refer to a complex contact manifold with a complex almost contact metric structure satisfying these conditions as a com-

plex contact metric manifold. For a given normalized contact structure we denote by A, as in the real case, the space of associated metrics. As a matter of notation we define local 2-forms G and H by G(X, Y) = g(X, GY) and H(X, Y) = g(X, HY). Bearing in mind that the local contact form 0 is u - iv to within a nonvanishing complex-valued function multiple and since in the overlap Oa fl Op,

up = au,, - bva, vQ = bu,, + avq, we can define an integral submanifold as a submanifold whose tangent spaces belong to the complex contact subbundle, i.e., u(X) = v(X) = 0 or equivalently 9(X) = 0. If the submanifold is itself a complex submanifold, we call it a holomorphic integral submanifold. When the holornorphic integral submanifold has complex dimension 1, it is called a holomorphic Legendre curve. Recall that in real contact geometry the maximum dimension of an integral submanifold of a contact manifold of dimension 2n + 1 is only n. Similarly since U and V are normal, from the above equations we see that for any integral submanifold, GX and HX are normal for any tangent vector X. Thus an integral submanifold of a complex contact manifold of complex dimension 2n + 1 has real dimension at most 2n.

seen in the earlier chapters that for a contact metric structure (NowWerlforhave g), the tensor field h defined by h = 2,Cgql plays a fundamental role. a complex contact metric structure we define local tensor fields by hu = 1 sym(GuG) o p,

by = 2 sym(GvH) o p

12.2 Examples of complex contact manifolds

193

where sym denotes the symmetric part. hu anti-commutes with G, by anticommutes with H and

VXV = -HX - HhvX - o-(X)U.

VXU = -GX - GhuX + o (X)V,

From these equations one readily sees that the integral surfaces of V are totally geodesic submanifolds. Just as the meaning of the vanishing of h in the real case is that the metric was invariant under the action of , i.e., is Killing, an associated metric g is projectable with respect to the foliation induced by the integrable subbundle V if and only if hu and by vanish.

12.2 12.2.1

Examples of complex contact manifolds Complex Heisenberg group

We have seen that 1183 with the Darboux form 77 = 2 (dz - ydx) as its contact structure and Sasakian metric g = 1(dx2 + d y2) + q 077 is a standard example

of a contact metric manifold. Identifying R3 with the Heisenberg group

HH=

1

y

z

0 0

1

x

0

1

x,y,zEIR

^1183

left translation preserves rl, and g is a left invariant metric on HR (see Example 4.5.1). The complex Heisenberg group is the closed subgroup He of GL(3, C) given by

He =

1

Z2

z3

0

1

0

0

z1 1

z1,z2,z3 E C

(C3.

If LB denotes left translation by B, LBdz1 = dz1 i LBdz2 = dz2, LB (dz3 z2dz1) = dz3 - z2dz1. The vector fields a + z2 3 , aa2 , aa3 are dual to the 1-forms dz1, dz2i dz3 - z2dz1 and are left invariant vector fields. Moreover relative to the coordinates (zl, z2, z3, zl, z2, z3) the Hermitian metric (Jayne [1992, p. 234])

0 1 + (z212

0

-z2

0

1

0

-z2

0

1

1 + Iz212

0

-z2

0

1

0

-z2

0

1

0

I

194

12. Complex Contact Manifolds

is a left invariant metric on Hc, but it is not a Kahler metric. The form

8=

2(dz3 - z2dz1) is a complex contact structure on Ho and in our view (He, 8, g) plays the role in the geometry of complex contact manifolds, that 1[83 with its standard Sasakian structure does in the geometry of real contact manifolds.

As we have seen, a complex contact manifold admits a complex almost contact structure. Here He -_ C3 and 8 is global, so the structure tensors may

be taken globally. With J denoting the standard almost complex structure on C3, J-x = ay, , we may give a complex almost contact structure to He as follows. Since 9 is holomorphic, setting 9 = u - iv, v = u o J. Also set 4az3 = U + iV; then u(X) = g(U, X) and v(X) = g(V, X). Finally in complex coordinates G and H were given by Jayne [1992, p. 235].

0

0

G=

,

H= 0

0

Note that the matrix gG is that of the real part d8 and the matrix gH is that of the imaginary part of dB. For this structure the 1-form o, vanishes. Now let

r=

1

`Y2

`Y3

0 0

1

-Y,

0

1

7k = Mk + ink, mk, nk E Z

;

r is a subgroup of HH _- V. The 1-form dz3 - z2dz1 is invariant under the action of r and hence the quotient He/r is a compact complex contact manifold with a global complex contact form. He/r is known as the Iwasawa manifold. The Iwasawa manifold has no Kahler structure, but it does have an indefinite Kahler structure and it has symplectic forms, see Fernandez and Gray [1986]. 12.2.2

Odd-dimensional complex projective space

We will need a local expression for the complex contact structure on GP2n+l; we will use homogeneous coordinates (z1i ... , zn+1 w 1 , . , wn+l) Then the complex contact structure is given by the holomorphic 1-form .

n+1

(zkdwk - WkdZk). k=1

.

12.2 Examples of complex contact manifolds

195

To give a little more detail we remark that the complex contact structure on CP2n+1 is closely related to the Sasakian 3-structure on the sphere S4n+3

(see Chapter 13) and to the quaternionic Kahler structure on quaternionic Hn+1 has three almost complex structures I, J, K which act on the position vector x as projective space. C2n+2

Ix = ix = (izl,... iz,,,+1, iwl,... ,

Jx = (i47v1,

...

Kx = (u11,

, i2Un+1,

,

iwT,,+1),

-iii,... , -i,zn+1),

... , wn+1, -zl, ...

,

-Zn+1)

The vector fields on S4n+3 given by Sl = -Ix, E2 = -Jx, 3 = -Kx are the characteristic vector fields of the three contact structures 771, 772, 773 on In terms of the complex coordinates, 771, 772, 773 on S4n+3 are the restrictions of the following forms on C2n+2 which we denote by the same letters. S4n+3.

. n+1

zkdzk - zkdzk +WkdWk - wkdwk), k=1

n+1

(zkdwk - Wkdzk).

b = 773 + 2172 =

k=1

Ishihara and Konishi [1979] proved that if one of the contact structures of a manifold M4n+3 with a Sasakian 3-structure (see Chapter 13) is regular, the base manifold M of the induced fibration is a complex contact manifold. The structure is constructed as follows. If (01, S1, 771, g), is the regular Sasakian

structure, then 01 and g are projectable. Let it denote the horizontal lift with respect to the principal S' bundle connection defined by 7ql. Then J defined and the projected metric form a Kahler structure on M by JX = (cf. Example 6.7.2). For a coordinate neighborhood U on M and a local cross section T of M4n}3 over U, 1-forms u and v and a tensor field G defined on U by

u(X) O 7r /= 72(T*X),

V(X) 0 7r = 773(T-X ),

GX = 7r. (027*X - 771 (T*X)S3 + 773 (7-X)6)

define the complex contact and complex almost contact structures on M. In the case of the Hopf fibration this is the standard Kahler structure on CP2n+1 with the Fubini-Study metric. With the Hopf fibration induced by l, 0 = 773+i772 is a local expression for the complex contact structure on CP2n+1

196

12. Complex Contact Manifolds

12.2.3

Twistor spaces

Generalizing the previous example we discuss twistor spaces over quaternionic

Kahler manifolds, the twistor space of quaternionic projective space being odd-dimensional complex projective space. Consider a Riemannian manifold (M4", g) whose holonomy group is contained in Sp(n) Sp(1) = Sp(n) x Sp(1)/{+I}). This means that there exists a subbundle E C End(TM) with 3-dimensional fibres such that locally there exists a basis of E consisting

of almost complex structures {T, J, JC} satisfying, TJ = -,71 = K and V xT, VxJ, V. K, belong to the span of IT, J, K } for any vector field X on M. In dimension 4 (n = 1) Sp(1) - Sp(l) = SO(4) so the holonomy group condition is not a restriction. For n > 1, (M4", g) is called a quaternionic Kdhler manifold. A well-known result of Alekseevskii [1968] (see also Berger [1966], Ishihara [1974]) is that in this case, the metric g is Einstein. Since Sp(1) Sp(l) = SO(4), a stronger definition of quaternionic Kahler

manifold is needed in dimension 4. A 4-dimensional manifold is said to be a quaternionic Kahler manifold if it is Einstein and self-dual with non-zero scalar curvature (see e.g., LeBrun [1991]). A 4-dimensional self-dual manifold is sometimes said to be half-conformally flat. Returning to the subbundle 7r : E --1 M, regardless of dimension, we define

the horizontal subspace '1-l, at p E E as follows. Let a denote a curve in M such that a(0) = Tr(p) and let s denote a section of E such that s(a(O)) = p. Then set RP = {(s o a)*(0) E TTE I Va(o) s = 0}. Now induce a bundle metric on E by making each quaternionic Kahler frame {T, 3, K } an orthonormal basis. Let Z be the space of all unit elements of E. Locally

Z={xI+y,7+z1CEEIx2+y2+z2=1}. : Z ---> M is called the twistor space of M and each element j E Z is an almost complex structure on the tangent space of M at 7r(j). Let V = ker 7r*. Since each fibre of Z is a unit sphere, then at j = xT + y,7 + z)C, x2 + y2 + z2 = 1, we may make the identification

7r

V, _ {X E E,(;) X 1 j } _ {al + b,7 + c1C I ax + by + cz = 0}. Setting 7-1 = RJz, we have the splitting TM = V ® H. Define an almost complex structure J on Z as follows. For a vertical vector V E Vi, set JV = j x V where x denotes the usual vector product in Euclidean 3-space; in particular this is the usual almost complex structure on S2. To define the action of J on X E 'H first let -denote the horizontal lift with respect to the connection on E determined by ?-l. Then since j is an almost complex structure on the tangent space of M at 7r(j), set JX = jnr,,X. Now local V-valued forms

12.2 Examples of complex contact manifolds

197

9 defining H then give Z a complex contact structure, see Salamon [1982] or for verification of the integrability Besse [1987, pp. 413-415] and for 9 A(d9)- 0, Besse [1987, p. 416].

Now let {U, } be an open covering of the quaternionic Kahler manifold (M4n, g) and corresponding to a neighborhood Uc. set

0a={xZ+y,7+zKEZIz

11, 0a={xZ+yj+z1CEZI z

-11.

The collection of pairs {O, Oa} is then an atlas on Z. On O,, define vector fields U and V by _ _ 2

U=

1

z lz zx Z- 1 xy,7+x1C

and -JU. Then {U, V } forms a basis of V on O. As elements of E, U and V are unit and there exist real 1-forms u and 13 such that 0 =,a ®U + v ® V and 13(U) = v(V) = 1, fi(V) = v(U) = 0. On O' define vector fields U' and V' by

U,- +z-x2- xy 1+z

and

1+z`7+x1C

-JU'. Again we have the corresponding 1-forms ic', v' and

U + it = h(U' + ii '),

fi - iv = h(u - iv')

where h : O. n Oa - + S'. Thus {13 - iv} is a normalized contact structure on Z and u ® u + v ®13 is a global tensor field on Z. Let c > 0 and define a Riemannian metric g, on Z by gC=c2(13®13+13(9 13)+7r*g.

g.;Iv = c2 where is the Euclidean metric on fibres of E restricted to S2. g, is called the Salamon-Berard-Bergery metric with vertical coefficient c.

Setting u, = cu, v, = cf, {u, - iv,} is a normalized contact structure on Z and

gr=u,®u,+vc®v.+7r g. In [1996] Foreman proved the following theorem.

Theorem 12.1 Let (M4ri, g) be a quaternionic Kahler manifold and T the scalar curvature of g. Consider the twistor space Z with the Salamon-BerardBergery metric g, c > 0 and normalized contact structure {u,, - ivC}. Then we have the following:

1. g, is Hermitian with respect to the complex structure J on Z. 2. g, is an associated metric if and only if clTl = 8n(n + 2). 3. g0 is Kahler if and only if c27- = 4n(n + 2). 4. g, is associated and Kahler if and only if c = 2 and T = 16n(n + 2).

198

12. Complex Contact Manifolds

Recall that in Section 1 we noted conditions for an associated metric g to be projectable with respect to the foliation induced by the integrable subbundle V, namely that hu and by vanish. Foreman [1996] also proves the following result.

Theorem 12.2 Let Z be the twistor space over a quaternionic Kahler manifold (M4n, g) of constant scalar curvature of r and let c be such that c17I = 8n(n + 2). Then in the space A of all associated metrics with respect to the normalized contact structure {uc - iv,}, the Salamon-Berard-Bergery metric gc is the only projectable associated metric in A. Again it seems worthwhile to mention the result of LeBrun [1995].

Theorem 12.3 A complex contact manifold of positive first Chern class is a twistor space if and only if it admits a Kahler-Einstein metric. Foreman [2000b] also gives curvature conditions for a complex contact manifold to be the twistor space of a quaternionic Kahler manifold; these conditions are conditions on the curvatures R X y U and RX yV . 12.2.1

The complex Boothby-Wang fibration

In [2000a] and again in [2000b] Foreman constructed complex contact manifolds with global complex contact forms and fibrations with vertical fibres Sl x S' and hence these examples are quite different from the twistor space examples. Here we discuss the complex Boothby-Wang fibration of Foreman [2000a]. Let (M, S2) be a complex symplectic manifold of complex dimension 2n and complex structure J0, i.e., M is a complex manifold together with a closed holomorphic 2-form S2 such that Sin # 0. Writing S2 = 521 + ic2 we see that S21 and i2 are closed 2-forms. On the other hand S2 may be written in terms of a local basis of holomorphic 1-forms as 01 A 0n+1 +

+ Bn A e2,,,. Then

taking real and imaginary parts of 6k = ak + i/3k we have

a2n-Nnn02n,

ci2=al AXi1+01Aan+1+"-+'ann02n-X na2n from which we see that Stirs 0 and S22n ; 0. Thus we have two distinct symplectic structures on M. If each of these is of integral class, we have two principal circle bundles PI and P2 with contact (connection) forms rll and 772 as in Sections 3.3, 4.5.4 and 6.7.2 and each dgk = Stk. Finally let Sl and 2 be the characteristic vector fields of these contact structures. Define a principal S' x Sl-bundle P over M by P = Pl ® P2 and let p denote the projection map. For z E P, set VP = z. This defines a vector bundle V over P and by extending each l;k to be trivial on the other factor, we may I

12.2 Examples of complex contact manifolds

199

regard 1 and S2 as vector fields on P. Moreover the pair (771,'q2) defines a connection on P (see Foreman [2000b] for details). We denote the horizontal subspace determined by the connection at z by 'HP P and the horizontal lift

by por X. The bundle space P carries an almost complex structure J defined as follows. On V define J by lye and Jet = -j1, and for X E ?"6,P, set Since the Lie algebra s1 ®s1 is abelian [1 1;2] = 0 and hence JX = pJop*X [J, J] (1i l;2) = 0. Also it is easy to check that [l;k, k] = 0 and hence that i

.

[J, J] (k, X) = 0. Finally utilizing the integrability of J0 on M one can readily show that p* [J, J] (X, k) = 0. Therefore P is a complex manifold. We can now define a complex contact structure on the complex manifold P. First note that '92 = -rll o J where again by extending each rlk to be trivial on the other factor we regard rll and rl2 as 1-forms on P. Set 0 = 211 + ir72; then 9 (X + i JX) = 0 so that 9 is a holomorphic 1-form. Moreover since dr1k is the pullback of S2k, dO = p*Q and a straightforward computation shows that 0 A (d9)"` 0. Thus P becomes a complex contact manifold with a global complex contact form. We summarize the above construction in the following theorem. Theorem 12.4 Let M be a complex symplectic manifold with a complex symplectic form S2 = Q1+iSl2 such that both SZl and S12 determine integral classes. Then the S1 x S1-bundle defined by ([S21], [ l2]) E H2(M,Z) ® H2(M,Z) has an integrable complex structure and a complex contact structure given by a holomorphic connection form, whose curvature form is Q. Foreman now proves a converse to this theorem as a complex Boothby-Wang fibration; we state the result here and refer to Foreman [2000b] for the proof. For a global complex contact form 0 we write 9 = u - iv where u and v are

real forms with v = u o J; the vertical bundle V is then spanned by global vector fields U and V = -JU where u(U) = 1,

v(U) = 0,

t(U)du = 0,

u(V) = 0,

v(V) = 1,

L(V)dv = 0.

Theorem 12.5 Let P be a (2n + 1)-dimensional compact complex contact manifold with a global form 9 = u - iv such that the corresponding vertical vector fields U and JU are regular. Then 9 generates a free Si x S1-action on P and p : P --> M is a principal S' x S'-bundle over a complex symplectic manifold M such that 9 is a connection form for this fibration and the complex symplectic form S2 on M is given by p* Q = d9.

200

12. Complex Contact Manifolds

12.2.5 S-dimensional homogeneous examples

In the previous example B. Foreman gave a complex Boothby-Wang fibration for (regular) complex contact manifold with a global contact form. In [1999] Foreman studied 3-dimensional complex homogeneous complex contact manifolds with a global complex contact form and obtained the following classification.

Theorem 12.6 If M is a 3-dimensional complex homogeneous complex contact manifold with global complex contact form, then M is of the form M = G/F where G is a simply connected 3-dimensional complex Lie group and F C G is a discrete subgroup. 1. Suppose G is unimodular. Then G is one of the following: (a) SL(2, (C), if rk(ad(V)) = 2.

(b) The universal cover of the group of rigid motions of the complex Euclidean plane, if rk(ad(V)) = 1.

(c) Hc, if rk(ad(V)) = 0. 2. Suppose G is not unimodular. Then G is solvable; rk(ad(V)) = 1; and G is one of the following complex Lie groups:

(a) The semi-direct product G, = C xT, C2, for any a E C*\1, where Tis a certain representation of C in GL(2, (C).

(b)G=

et 0 0

tet u et 0

v

t,u,vE(C

1

12.2.6 Cn+1 X CPn(16)

Korkmaz [1998] gives a complex analogue of the real contact metric manifold En+1 x Sn(4) together with a curvature characterization analogous to Theorem 7.5. We describe the example and state the theorem. Let (to,... , tn) be homogeneous coordinates on CPn and ui the neighborhood defined by ti 0. On ui introduce non-homogeneous coordinates by wj = ty , j = 0, ... , n, j i. Let Oi = Cn+1 x ui and let (zo, ... , zn) be coordinates on Cn+1 Define a holomorphic 1-form Oi on Oi by Bi =

1 ti

tkdzk. k=O

0 on Oi and Oj = t Bi on Oi n Off. Thus {9j} o is a complex contact structure on Cn+1 X CPn. Then Bi n (d9i)n

12.2 Examples of complex contact manifolds

201

For convenience we continue our work on Oo with 00 = dz° + Ek=0 wkdzk. The product metric on (Cn+1 x CPn(16) is given by the matrix

0\

In+1

0

g

0

98

1n+1

0

0

gT

\

0

where $g is the metric on (CPn(16), i.e.,

_

gij

(1 + Ek_1 lwkl2)a.7 w2wj Fn lwkl2)2 (1 + k=1

Let fo = 1 + Ek=1 lwkl2 and define real 1-forms u0 and vo by uo =

4

dz° + dz° + I

v° =

(dz°

4

(wkdzk + wkdzk)

,

k=1

- dz° +

\

n

(wkdzk

- wkdzk)

k=1

Similarly define vector fields UO and VO by 2

r a

U0\az°+az° _

-2iG-zo a

a az°

VO

Then 00 = 2

a

a

(t v-

azk

k=1

k

k=1

a

(w azk

a + w k azk) I ,

a I - w kazk)

.

(uo - iv0), duo(Uo,X) = 0 for all X E 71, uo(Uo) = 1,

vo(Uo) = 0 and g(Uo,X) = uo(X) for all X. Let 0

A\

B

0

0 Go = 0

A

0 0

202

12. Complex Contact Manifolds

where w2

wl 112 1W

A=

- fo

...

2131w2 212

wlw2

1W

- fo ...

W22Un

wn

wlwn

...

2U2wn

IwnI2

-w2 B = f2

- fo J

Then Go = -I + uo ® Uo + vo ® Vo, GoJ + JGo = 0, etc. On 01 set fl = ,12 1 + Iw° IZ + Ek=2 Iwk I2. Then f = t . Setting a - bi = f° t? on Go n 01 we have a2 + b2 = 1 and

ul = auo - bvo,

vl = buo + avo,

Gl = aGo - bHo,

Hl = bGo + alo

where ul, v1, G1, Hl are the structure tensors on 01. Therefore n J/(nk, vk, Uk, Vk, Gk, Hk, 9) 1 on Ok k=0

is a complex contact metric structure on Cn+1 x (CP7'(16).

It can be shown that for this structure hu = hv, see Korkmaz [1998]. We now state a characterization of this example, due to Korkmaz [1998], and analogous to Theorem 7.5, that a contact metric manifold on which RXy = 0 is locally isometric to En+1 X Sn(4).

Theorem 12.7 Let M be a complex contact metric manifold with hu = hv. If RXyV = 0, then M is locally isometric to Cn+1 X (CPn(16).

12.3

Normality of complex contact manifolds

As we have seen in real contact geometry, the product M x R of an almost contact manifold and the real line carries a natural almost complex structure J and the almost contact structure is said to be normal, if J is integrable. Recall also that a Sasakian manifold is a normal contact metric manifold. Ishihara and Konishi [1979], [1980] introduced a notion of normality for complex contact structures. Their notion is the vanishing of the two tensor fields S and T given by S(X, Y) = [G, G] (X, Y) + 26(X, Y)U - 2H(X, Y)V + 2(v(Y)HX - v(X)HY)

+a(GY)HX - a(GX)HY + a(X)GHY - a(Y)GHX, T(X,Y) = [H,H](X,Y)-26(X,Y)U+2H(X,Y)V+2(u(Y)GX-u(X)GY) +a(HX)GY - a(HY)GX + a(X)GHY - a(Y)GHX.

12.3 Normality of complex contact manifolds

203

However this notion seems to be too strong; among its implications is that the underlying Hermitian manifold (M, g) is Kahler. Thus while indeed one of the canonical examples of a complex contact manifold, the odd-dimensional

complex projective space, is normal in this sense, the complex Heisenberg group, is not. B. Korkmaz [2000] generalized the notion of normality and we adopt her definition here. A complex contact metric structure is normal if S(X, Y) = T (X, Y) = 0, for every X, Y E 'H,

S(U, X) = T (V, X) = 0, for every X.

Even though the definition appears to depend on the special nature of U and V, it respects the change in overlaps, O. n Op, and is a global notion (Korkmaz [2000]). With this notion of normality both odd-dimensional complex projective space and the complex Heisenberg group with their standard complex contact metric structures are normal. We remark also that for Ishihara and Konishi's notion of normality, the holonomy group is the full unitary group U(2m + 1), Houh [1976].

One consequence of normality is that hu = 0 for every U E V; this is analogous to the fact that every Sasakian structure is K-contact. Another is that the sectional curvature of a plane section spanned by a vector in V and a vector in 7-l is equal to +1 (cf. Korkmaz [2000], Foreman [2000b]). We now give expressions for the covariant derivatives of G and H on a normal complex contact metric manifold; for proofs see Korkmaz [2000]. Setting SZ = do- we have that a complex contact metric manifold is normal if and only if the covariant derivatives of G and H have the following forms.

g((VXG)Y, Z) = 2a(X)g(HY, Z) + 2v(X)1l(GZ, GY) - 2v(X)g(HGY, Z)

-u(Y)g(X, Z) - v(Y)g(JX, Z) + u(Z)g(X, Y) + v(Z)g(JX, Y). g((VxH)Y, Z) = -2o(X)g(GY, Z) - 2u(X)1(HZ, HY) - 2u(X)g(GHY, Z) +u(Y)g(JX, Z) - v(Y)g(X, Z) + u(Z)g(X, JY) + v(Z)g(X, Y).

In these formulas the first two terms on the right vanish for the complex Heisenberg group (Example 12.2.1) and the second and third terms cancel on PC2n+' (Example 12.2.2). Also one has

VXU = -GX, VXV = -HX on He and

VXU = -GX + o(X)V, VxV = -HX - c(X)U

204

12. Complex Contact Manifolds

on CP2,+i Finally on a normal complex contact manifold g((VxJ)Y, Z)

= 2u(X)(s2(Z,GY) - g(HY, Z)) + 2v(X)(S2(Z, HY) + g(GY, Z)). When the complex contact structure is strict, i.e., given by a global complex contact form, the situation is more restrictive. In particular a = 0 and therefore some of the above formulas simplify. Foreman [2000b] defines a complex Sasakian manifold to be a normal complex contact metric manifold whose complex contact structure is given by a global complex contact form. His paper gives a number of basic properties and examples including local projectivity to a hyper-Kahler manifold.

Finally we mention that in [toap] Korkmaz develops a theory of complex (ic, µ)-spaces analogous to that described in Section 7.2.

12.4

GH-sectional curvature

Corresponding to the ideas of holomorphic curvature in complex geometry and ql-sectional curvature in real contact geometry, B. Korkmaz [2000] defined the notion of GH- sectional curvature for a complex contact metric manifold. For

a unit vector X E '-im, the plane in TmM spanned by X and Y = aGX + bHX, a, b E R, a2 + b2 = 1 is called a GH-plane section and its sectional curvature, K(X, Y), the GH-sectional curvature of the plane section. For a given vector X, K(X, Y) is independent of the vector Y in the plane of GX and

HX if and only if K(X, GX) = K(X, HX) and g(RxGxHX, X) = 0. Let M be a normal complex contact metric manifold; if the GH-sectional curvature is independent of the choice of GH-section at each point, it is constant on the manifold and we say that M is a complex contact space form. The curvature tensor and the following theorems were obtained by Korkmaz [2000]; explicitly the curvature tensor is 3

RxyZ = c

(g(Y, Z)X - g(X, Z)Y

4

+g(Z, JY)JX - g(Z, JX)JY + 2g(X, JY)JZ)

+c 41(- (u(Y)u(Z) + v(Y)v(Z))X + (u(X)u(Z) + v(X)v(Z))Y +2u A v(Z, Y)JX - 2u Av(Z,X)JY+4u Av(X,Y)JZ +g(Z, GY)GX - g(Z, GX)GY + 2g(X, GY)GZ

+g(Z, HY)HX - g(Z, HX)HY + 2g(X, HY)HZ

12.4 CH-sectional curvature

205

+( - u(X)g(Y, Z) + u(Y)g(X, Z) +v(X)g(JY, Z) - v(Y)g(JX, Z) + 2v(Z)g(X, JY)) U

+(

- v(X)g(Y, Z) + v(Y)g(X, Z)

-u(X)g(JY, Z) + u(Y)g(JX, Z) - 2u(Z)g(X, JY))V)

-

(52(U, V)+c+ l) ((v(X)unv(Z, Y) -v(Y)unv(Z, X)+2v(Z)unv (X, Y)) U 3

-(u(X)u A v(Z,Y) - u(Y)u A v(Z, X) + 2u(Z)u A v(X,Y))V).

Odd-dimensional complex projective space with the Fubini-Study metric of constant holomorphic curvature 4 is of constant GH-sectional curvature 1. The complex Heisenberg group has holomorphic curvature 0 for horizontal and vertical holomorphic sections and constant GH-sectional curvature -3.

Theorem 12.8 Let M be a normal complex contact metric manifold. Then M has constant GH-sectional curvature c, if and only if for X horizontal, the holomorphic sectional curvature of the plane spanned by X and JX is c + 3. Theorem 12.9 Let M be a normal complex contact metric manifold of constant GH-sectional curvature +1 and satisfying do (V, U) = 2, then M has constant holomorphic curvature 4. If, in addition, M is complete and simply connected, then M is isometric to CP2n+1 with the Fubini-Study metric of constant holomorphic curvature 4. Korkmaz [2000] then introduced the idea of an R-homothetic deformation of a complex contact metric structure. Let a be a positive constant and consider the local structure tensors (G, H, U, V, u, v, g). Then define new structure tensors by

u=au,

v=ctiv,

U= a-U, V= 1V, G=G, H=H, a ag+a(a-1)(u(9 u+v®v).

This change of structure is called an f-homothetic deformation. The new structure then respects the transitions on the overlaps of coordinate neighborhoods and hence gives a new complex contact metric structure on M. More-

over S(X,Y) = S(X,Y) on 'h, S(X, U) = 1 S(X, U), etc., so if the given

a

structure is normal, so is the new structure. Korkmaz computed the curvature and showed that if on a normal complex contact metric manifold the original structure has constant GH-sectional curvature c, then the new structure has constant GH-sectional curvature c = c + 3 - 3; in particular she proved the following results.

a

206

12. Complex Contact Manifolds

Theorem 12.10 Complex projective space CP2,+i carries a normal complex 4 contact metric structure with constant GH-section curvature - - 3 for every

a

a>0.

Theorem 12.11 A normal complex contact metric manifold with metric j of constant GH-sectional curvature c > -3 is H-homothetic to a normal complex contact metric manifold with metric g of constant GH-section curvature c = 1. 2

Moreover, if du(V, U) = (c 83) holomorphic curvature 4. 12.5

,

the metric g is Kahler and has constant

The set of associated metrics and integral functionals

In [1996] B. Foreman began the study of the set of all associated metrics on a complex contact manifold. In Chapter 4 we studied how the set of associated metrics A for a symplectic or contact form sits in the set of Riemannian metrics with the same volume element and in particular we studied the tangent space to A at an associated metric g. As before we will typically use the same letter

for a symmetric tensor field as a tangent vector and for the corresponding tensor field of type (1,1) determined by the metric. As already mentioned, and in keeping with the notation in the real case, we will denote by A the set of all associated metrics for a complex contact structure as defined in Section 12.1. As in the real case, A is infinite dimensional; all associated metrics to a given complex contact structure have the same volume element; A is totally geodesic inN in the sense that if D e T9A, the geodesic gt = getD is a path in A; and two metrics in A may be joined by a geodesic. We now give a characterization of T9A; for details of this and the preceeding statements see Foreman [1996].

Lemma 12.1 Let g be a metric in A. Then a symmetric tensor field D of type (0, 2) is in T9A if and only if D, as a tensor field of type (1, 1), satisfies

DJ=JD, Dw=0, DG=-GD for any local tensor field G as above. As before, to study integral functionals defined on A we will be differentiating such functionals along paths of metrics and for this we need the following fundamental lemma (Foreman [1996]). For an endomorphism T of a complex vector space, set

T3= 2(T-JTJ), Td= 2(T+JTJ),

12.5 The set of associated metrics and integral functionals

207

i.e., TS is the part of T that commutes with J and Td the part of T that anti-commutes with J.

Lemma 12.2 For g E A and T a symmetric (1, 1) tensor field, trTD dV = 0 fM

for every D E T9A if and only if

p(TJ+JT)p=HTG-GTH or equivalently

pTsp = -GT3G. In Section 10.3 we studied the integral of the Ricci curvature in the direction

of the characteristic vector field , i.e., L(g) =

cRic(1)

'M

dV, as a functional

on the set of associated metrics. In [1996] Foreman gave two analogues of Ric(h) for complex contact manifolds and proved the following results. For a unit vertical vector field U, Ric(U) + Ric(JU) is global, i.e., respects transition on Oa n Op, and we denote it by Ric(V). For the second analogue, Foreman utilizes the *-Ricci tensor and it is easy to check that for a unit vertical vector is independent of the unit vertical vector field U field U, Ric*(U) = and globally defined; we denote it by Ric* (V). Define functionals L and L* on A by

L(g) =

JM

Ric(V) dV,

L*(g) = f Ric*(V) dV. M

Theorem 12.12 Let M be a compact complex contact manifold and A the set of associated metrics. Then g E A is a critical point of L(g) if and only if

(Vuhu)3 + (Vvhv)3 = a(U)h' - o (V)hj + 4kuhd for any local unit vertical vector field U.

Clearly any projectable associated metric is a critical point of L(g). In fact Foreman showed that Ric(V) = 8n - 2K(V) - trh' - trh2 where K(V) is the sectional curvature of a vertical plane section and moreover that f K(V) dV M

is independent of the associated metric. Thus projectable metrics are maxima of L(g). If g is Kahler, hu = 0 for any vertical vector field U and moreover (Vwh r)9 = 0 for any vertical vector field W. Thus we also have the following corollary.

Corollary 12.1 If g E A is Kahler and a critical point of L(g), then it is projectable.

208

12. Complex Contact Manifolds

Theorem 12.13 Let M be a compact complex contact manifold and A the set of associated metrics. Then g E A is a critical point of L*(g) if and only if hd (V U J) = 0

for any local unit vertical vector field U. In particular, if g is projectable or Kahler, it is critical. In complex dimension 3, this theorem may be improved to g being critical if and only if locally either hU = 0 or J is parallel along V, i.e., VuJ = 0 for any local unit vertical vector field U. For the twistor spaces as described in Example 12.2.3 this theorem may be improved as follows. If Z is the twister space of a compact quaternionic Kahler manifold (M4n, g), 4n > 8, with normalized contact structure {u,-ivc} and if the Salamon-Berard-Bergery metric g, is an associated metric, then an associated metric g is critical for L* if and only if hU = 0 for every vertical vector U. B. Foreman also obtained some important results on the constancy of L*, i.e., when L* is independent of the associated metric.

Theorem 12.14 Let M be a complex contact manifold with corresponding almost complex structure J. Then L* is constant on A if and only if J is projectable.

Theorem 12.15 Suppose M is a compact complex contact manifold with a global complex contact structure and A the set of associated metrics. Then L* is constant on A.

Turning now to the scalar curvatures, in real contact geometry the "total scalar curvature" defined by f T + T* dV is a functional on A whose m

critical points are precisely the K-contact metrics as we have seen (Theorem 10.9). B. Foreman (unpublished) has also defined a **-scalar curvature by first contracting the curvature with the local tensor fields G and H as in the *-scalar curvature giving *-scalar curvatures, TG*, and r ; T** = TG + TH is then globally defined and one defines the "total scalar curvature" 1(g) by 1(g) = f T + T* + T** dV. Recall that an almost Hermitian structure (J, g) M

is semi-Kahler if the fundamental 2-form Sl is co-closed. Foreman computed the critical point condition for this functional and, his computation, though complicated, yields the following result.

Theorem 12.16 A projectable semi-Kdhler metric is a critical point of the functional 1(g).

12.6 Holomorphic Legendre curves

12.6

209

Holomorphic Legendre curves

In Section 8.3 we saw that in a 3-dimensional Sasakian manifold, Legendre curves are characterized by their torsion being equal to 1 and initial conditions at one point and that without the initial conditions, a counterexample can be given. Moreover, in R3 with its standard Sasakian structure the curvature of a Legendre curve is twice the curvature of its projection to the xy-plane with respect to the Euclidean metric. Our first purpose here is to study the complex Heisenberg group Hc as a complex contact manifold and to prove similar results for this space. One important difference from the real case is that while holomorphic Legendre curves in He have torsion equal to 1, this is the only possible non-zero constant. In fact the first main result here is that a holomorphic Frenet curve in He whose torsion is not identically zero and satisfies an initial condition at one point is a holomorphic Legendre curve. Let M be a Hermitian manifold of complex dimension n with complex structure J and corresponding Riemannian metric g. Following Chern, Cowen and Vitter [1974], S. Dolbeault [1977] we describe holomorphic curves and Frenet frames. A holomorphic curve in M is a non-constant holomorphic map : M -+ M, where M is a Riemann surface. If zi = zi(w) is a local representation of M in a neighborhood of w = 0, then its holomorphic tangent vector at a(0) is given by E zi(0) a. = wPV where p is a non-negative integer and V a non-zero vector. The isolated points where p > 0 are called stationary points of order 0. A unitary frame { f1 ... fn} is called a Frenet frame if f1 = IVvI L.

and

VX fi = Wii-1(X) fi-1 + Wii (X) fi + wii+1(X) fi+1

for i = 1 to n - 1 where wii+1 is a holomorphic 1-form and wi+li(X) _ -wii+1(X). Points where wii+1 vanish are called stationary points of order i. In general a unitary frame is not holomorphic but we will be interested in the case where the fi's are holomorphic vector fields and we then speak of a holomorphic Frenet curve and a holomorphic Frenet frame. Not every holomorphic curve in a Hermitian manifold M has a Frenet frame. Chern, Cowen and Vitter [1974] (or see Dolbeault [1977]) give curvature con-

ditions on M under which every holomorphic curve has a Frenet frame; in particular a Kdhler manifold of complex dimension > 3 has a Frenet frame along every holomorphic curve if and only if it has constant holomorphic curvature. M being a holomorphic Frenet curve has implications on M as a submanifold. In the following lemma we give three such implications, the first two of which are immediate in any Kahler manifold. For our purpose we give the lemma only for Hermitian manifolds of complex dimension 3 (see Baikoussis, Blair, and Gouli-Andreou [1998] for the proof). Recall that the span of the

210

12. Complex Contact Manifolds

second fundamental form a is called the first normal space and will be denoted by vl. Let `proj' denote projection to the orthogonal complement of TM ®vl. Define ,(3(X, Y, Z) by

a(X,Y, Z) = projVxVYZ(= projV cl(1', Z)). The span of 0 is called the second normal space and will be denoted by v2. Successively the higher normal spaces may be defined in this manner taking higher order derivatives. Lemma 12.3 Let M be a holomorphic Frenet curve in a 3-dimensional Hermitian manifold (M, J, g) with second fundamental form a. Let R denote the curvature tensor of M. Then

1) a(X, JY) = Ja(X, Y) for any tangent vectors X, Y and hence the real dimension of the first normal space is 2, 2) VXJITM®T1M = 0,

3) for any tangent vectors X, Y, Z, we have that R(X, Y)Z is orthogonal to the second normal space.

Conversely if M is a holomorphic curve satisfying 1), 2) and 3), then it has a holomorphic Frenet frame. Related to the idea of a Frenet frame are the curvatures themselves. These Frenet curvatures for a holomorphic curve date back to Calabi [1953a] who defined (n - 1) real-valued curvature functions (see also Lawson [1970]). When

the ambient space is a complex space form these curvatures are actually intrinsic (Lawson [1970]). We follow the development as presented by Lawson [1970].

Let M be a holomorphic curve in a 3-dimensional Hermitian manifold M for

which the properties of the lemma hold. From a(X, JY) = Ja(X,Y) we have easily that for all unit tangent vectors X, Y (Y not necessarily distinct from X), a(X, Y) I2 is a function of position alone, say c1(p), p E M; tt1 is called the curvature or first curvature of M. Now with (X, Y, Z) defined as above, from property 2) of the lemma we have O(X, Y, JZ) = J,3(X, Y, Z). From the second expression for 0 in the definition we see that /3 is symmetric in the second and

third variables giving /3(X, JY, Z) = J,6(X,Y, Z). Let v be a vector in the second normal space, then by property 3), g(VxVyZ, v) = g(VyVxZ, v), giving that 8 is symmetric in the first and second variables and therefore i3(JX, Y, Z) = J/3(X, Y, Z). From these properties of /3 we have that K2 (p) -

IP(X,Y,Z)12 /cl (p)

12.6 Holomorphic Legendre curves

211

is well defined, X, Y, Z being any unit tangent vectors. We call Ic2 the torsion or second curvature and also denote it by r. It is interesting to compare the curvature and torsion with the derivatives of the Frenet frames and the holomorphic connection forms wii+1. In particular

writing fl as

ei-Jei

K = Iw12(el)I2 and r = IW23(e1)I2.

Now analogous to the starting point in the real case, Proposition 8.2, we begin with the following proposition from Baikoussis, Blair, and Gouli-Andreou [1998].

Proposition 12.1 Let M be a real surface in (Hc, 9, g) such that 0(X) = 0 for any tangent vector X. Then M is a holomorphic Legendre curve as well as a holomorphic Frenet curve with torsion r - 1. Our first result, due to Baikoussis, Blair, and Gouli-Andreou [1998], is the complex analogue of Theorem 8.12. Theorem 12.17 If the torsion of a holomorphic Frenet curve in the complex Heisenberg group is not identically zero and at one point the complex contact form annihilates the tangent space, then the curve is Legendre. Corresponding to Theorem 8.14 we also have the following result (Baikoussis, Blair, and Gouli-Andreou [1998]).

Theorem 12.18 Let M be a holomorphic Legendre curve in (Hc, 9, g) and N its projection to C2 = {(z1, z2)} with its standard complex structure and Kdhler (Euclidean) metric. Then the Gaussian curvature of M is 8 times that of N. Turning to CP3 we give a description of holomorphic Legendre curves due to Bryant [1982].

Theorem 12.19 For any compact Riemann surface there exists a holomorphic Legendre imbedding into CP3.

These holomorphic Legendre curves in CP3 can be described as follows. Let (z1i z2, w1, w2) be homogeneous coordinates on CP3 and 0 = z1dw1- w1dz1 + z2dw2 - w2dz2 as in Example 12.2.2. Now for a connected Riemann surface M, let f and g be meromorphic functions on M with g non-constant. Define

c:M --f CP3 by

f' f'

-' (1, g, f -g2g,, 2g,), then it is easy to check that 2b(t*&) = 0. Conversely Bryant shows that any holomorphic Legendre curve L : M -* CP3 is either of this form or has its image in some CP' C CP3. As an application of his result Bryant uses the fibration of CP3 over S4 to show that given a compact Riemann surface,

212

12. Complex Contact Manifolds

there exists a conformal superminimal generically 1-1 immersion into S4 whose image in S4 is an algebraic surface.

12.7

The Calabi (Veronese) imbeddings as integral submanifolds of CP2n+l

In [1953b] Calabi showed that up to holomorphic congruence there is a unique holomorphic imbedding of CPn(U) into cCPN(4), N 1 which does not lie in any totally geodesic complex projective space of lower dimension. Nakagawa and Ogiue [1976] showed that the only full isometric immersions of positively curved complex space forms into positively curved complex space forms are local versions of these imbeddings. These imbeddings are known as the Calabi imbeddings or as the Veronese imbeddings, especially in the case

CP2 (2) -f CPs (4) For n = 1 these imbeddings are called Calabi curves. .

Classically the Calabi imbeddings are given as follows. Let S1, ... , Sn+l be homogeneous coordinates for CPn(v). The Calabi imbedding of CPn(v) into CPN(4), in terms of homogeneous coordinates for CPN(4) is given by rr

/

(Sl> ... Sn+1)

(Q/,

)

r

r

r

an 1, ... a1 ...vIan+1 Sl l ... (n+1

r

, Sn+l )

l'

where En ai = v, the ai's being non-negative integers. The meaning of the integer v is that there are v - 1 normal spaces for these imbeddings. The question to be addressed in this section is the following. For which of these imbeddings is there a holomorphic congruence of CP2n+1 which positions the submanifold as a holomorphic integral submanifold of the complex contact

structure? For the Calabi curves we have the following positive answer due to Blair, Dillen, Verstraelen, and Vrancken [1996]. Theorem 12.20 There exists a holomorphic congruence of CP2n+1(4) which positions the Calabi curve CP'(2 ) as a holomorphic Legendre curve in CP2n+1(4).

The imbedding is given explicitly as follows. For simplicity set Ak =

we position CP1(2n+1) by a holomorphic congruence of (1) as

(S1, r2) (rr2n+1 S1 ,...

/222n+1 S

,. . .

,

(-

An+1S1 rrn+l rn

r2n+2-krk-1 A kb1 S2

S2

W- lAk //--2n+2-kcrk-1,

(

i

.

)

1 nAn

(-)

rrn+1n)sl

+lb2

(2n+1) .

k-1

12.7 The Calabi (Veronese) imbeddings

213

Then a

in

Ak(2n+2-k)c

1-k CPN(4) is an integral submanifold of the complex contact structure with second fundamental form a. We have already noted that the vector fields U and V are

normal and that for any tangent vector X, GX is normal. Using VXU = -GX + o (X)V we have 0 = Xg(Y, U) = g(OXY, U) - g(Y, GX) = g(a(X,Y), U)

and similarly g(a(X, Y), V) = 0. Thus U and V are orthogonal to both the tangent space and the first normal space, but the first normal space is the whole normal space, giving a contradiction.

For v = 3 we first give a non-existence result for n = 2 and then a positive result for n odd; these results are due to Blair, Korkmaz, and Vrancken [2000].

Theorem 12.22 There is no holomorphic congruence of CP9(4) that brings the Calabi imbedding of CP2(A) into position as an integral submanifold of the complex contact structure on CP9(4).

Theorem 12.23 When v = 3 and n is odd, N = (11133) - 1 is odd and there exists a holomorphic congruence of CPN(4) that brings the Calabi imbedding

of CPn(3) into position as an integral submanifold of the complex contact structure on CPN(4).

214

12. Complex Contact Manifolds

Finally we give an example with v = 5. In terms of the homogeneous coordinates (z1i ... ) z28i w1, ... , w28), CP3 (5) may be realized as an integral submanifold of the complex contact structure on CP55(4) in the following way:

z1=(1

wl=-(3s

Z2 = (2

w2 = -(4

z3=

Y5SIS2

w3

-V5-G(4

Z4

-

-rl(3

W4

v5(34(l

V

ZS

=V

r

5 (4

W5 = V5 (34

(1

W6

Z7 = -,/5 (2 (3

W7

Z6 =

S 11

/5-(r

-

z8 =

v/5 rS2 rS4

zg =

10 r(1

-

w8 =

r4(

5(4(3 5(451 V/-5(4 (52

r Wg=-10(3(4

Z10 =

10(1 (3

rya

W10 = - 10(3 (1

z11 =

1015(4

w11 = -1053(2

Z12 =

10(2 (1

W12=- 10 (54(3

Z13 =

10(x2 (3

W13 = -V/10(45

Z14 =

10 (2 (4

W14 =

z15 =

20(1(2(3

W15 =

r r 20(3(1(4

Z16 =

20(1((2((4

w16 =

20(3(2(4

Z17 =

20

(((1

(3(4

W17 =

12

- 1'---0 (4 (

2

- 2 0(

3 152

x

/ /

z18 =

20(2(1(3

W18 =

20(4(1(3

zlg =

20(2(1(4

wig =

20(4(2(3

Z20 =

20(2(3(4

W20 = - 20(4(1(2

Z21 = 301(2 (3

W21 =

30(3 (4 (1

30(1(2 (4

W22 =

30(3(4 (2

Z23 = v/30(12 (3(2

-

W23 =

- 30 3 (S1 /(4

Z24 =

30(1 2 (2

W24 =

/ / - 30(3(2(4

Z25 =

30(1(4(3

W25 =

30(3(2(1

Z26 =

30(2 4 (1

22 W26 = - 30(2(4(3

Z27 =

60(1(2(3(4

W27 = -N/-6-0-(%3J(2(4

Z28 =

60(2(1(3(4

W28 =

Z22 =

60(2(1(2(3

13 3-Sasakian Manifolds

13.1

3-Sasakian manifolds

As with the last chapter we will give more of a survey and only a few proofs. Another survey of both history and recent work on 3-Sasakian manifolds is Boyer and Galicki [1999].

If a manifold M2r.+1 admits three almost contact structures i= 1, 2, 3 satisfying the following for an even permutation (i, j, k) of (1, 2, 3), Ok = qliOj - 77j ®Si = -030i + 77i

Sk =OiSj = -0311,

77k=77i0`i = -773 0ci,

then the manifold is said to have an almost contact 3-structure. This notion was introduced by Kuo [1970] and independently under the name almost coquaternion structure by Udriste [1969]. Some authors follow different sign etc. (see e.g., the latter part of Secconventions, taking q1k = -Oiq1j +71j tion 13.2 and Baikoussis and Blair [1995]). Note that given two almost contact metric structures, satisfying C 0102-772®`1=-0201+771 0

2,

t _ 71(e2) = 0, /2(`1) there exists a third almost contact structure defined by 1t2t = -02e1,

771 0 02 = -772 O 01,

03=0102-772®S1, giving an almost contact 3-structure.

e3=01e2,

773 = -772 0 01

216

13. 3-Sasakian Manifolds

Now given an almost contact 3-structure (0i, i, 77i), define on M2-+l x JR three almost complex structures Ji using each of the almost contact structures as in Section 6.1. It is then easy to check that Jk = JiJ; _ -JjJi. Therefore M2-+l x R has an almost quaternionic structure and hence its dimension is a multiple of 4. Thus the dimension of a manifold with an almost contact 3-structure is of the form 4n + 3. Tachibana and Yu [1970] used this idea to show that there cannot be a fourth almost contact structure (4S4, 774) i = 1, 2, 3, and satisfying the anti-commutativity with conditions with the first three structures. To see this let J4 be the almost complex structure on M2-+1 x R constructed using (04, 4, 774) Then pairing J4 with each of J1, J2, J3 giving J4Ji = -JiJ4, i = 1, 2, 3. This contradicts J3 J4 = J1 J2 J4 = -J1J4J2 = J4 J1 J2 = J4 J3. The normality of these almost contact structures was discussed by Yano, Ishihara and Konishi [1973]. In particular if two of the almost contact structures are normal, then so is the third. Kuo [1970] proved that given an almost contact 3-structure, there exists a Riemannian metric compatible with each of then and hence we can speak of an almost contact metric 3-structure (¢i, i, r/ j, g), i = 1, 2, 3. He also showed that the structural group of the tangent bundle is reducible to Sp(n) x 13. Moreover the vector fields {i1i 2, 31 are orthonormal with respect to the compatible metric. If the three structures (qli, i, 77i, g) are contact metric structures we say that M4n+3 has a contact metric 3-structure. If the three structures are Sasakian, we say that M4n+3 has 3-Sasakian structure, also called a Sasakian 3-structure and M4n+s is a 3-Sasakian manifold. Similarly we could define, and will occasionally use, the term K-contact 3-structure; however Kashiwada [toap] recently proved the very remarkable result that every contact metric 3-structure is 3Sasakian and we now give the proof in detail. The crux of Kashiwada's proof is a generalization of a lemma of Hitchin [1987] to the effect that given three

almost complex structures Ji, i = 1, 2, 3, satisfying Jk = JiJ,j = -J; Ji and a metric g that is compatible with each of them, then if each of the three fundamental 2-forms is closed, each Ji is integrable. Kashiwada [1998] generalized this lemma as follows.

Lemma 13.1 Let M4n be a differentiable manifold admitting three almost complex structures Ji, i = 1, 2, 3, satisfying Jk = JjJ; = -J,Ji and a metric g that is compatible with each of them, and let S2i, i = 1, 2, 3, be the corresponding

fundamental 2-forms. If dSli = 2w A 52i, i = 1, 2, 3, for some 1-form w, then each Ji is integrable.

Proof. First observe that

J1(VzJ2)Jl = -J1VzJ3 - J3VzJ1 = -JiVZJ3 - VzJ2 + (VzJ3)Ji

13.1 3-Sasakian manifolds

217

and hence that

(V cl2)(J1X, J1Y) _ -9(X, J1(VzJ2)J1Y)

_ (VZQ2)(X,Y) - (Vzl3)(J1X,Y) - (Ozl3)(X,J1Y).

(*)

Now recall that the Nijenhuis tensor of J1 can be written

[J1, Ji](X,Y) _ (J1VYJ1 -Vj1YJ1)X - (J1VxJ1 -Vj1xJ1)Y.

Then J2[Ji, J1](X,Y) = -J3(VYJI)X -J2(Vj1YJ1)X+J3(VxJ1)Y+J2(VJ1xJi)Y and by straightforward computation g(J2[J1i J1] (X, Y), Z)

= -3dSl2(X,Y, Z) + (VzQ2)(X,Y)

+3dS23(X, J1Y, Z)

- (Vz23)(X, J1Y)

+3d.l3(J1X, Y, Z) - (Vzf 3)(J1X,Y) +3dSl2(J1X, J1Y, Z) - (VzS22)(J1X, J1Y). Equation (*) and dS2z = 2w A S2i then yield [J1, J1] = 0. Similarly one proves

that J2 and J3 are integrable. We now prove Kashiwada's theorem.

Theorem 13.1 Every contact metric 3-structure is 3-Sasakian. Proof. Let M4'2+3 be manifold with contact metric 3-structure 9i, g), i = 1,2,3. On M4"+3 x R1 define three almost Hermitian structures (Ji, G) as in Chapter 6, i.e.,

Ji(X,fd) _ (OiX

G=g+dt2.

Then Jk = Ji Jj = - J, Ji and the fundamental 2-forms Sti satisfy dS l = 2w A Sti

where w = -dt. Now by Lemma 13.1 each Ji is integrable and hence each contact metric structure (0j, j, rti, g) is Sasakian. As we remarked at the end of Section 6.5, Boyer and Galicki [1999] define a 3-Sasakian manifold by considering the cone R+ x M. A Riemannian manifold

(Mm, g) is a 3-Sasakian manifold if and only if the holonomy group of the cone ([8+ x M, dr2 +r2g) reduces to a subgroup of Sp(' -4+1). Again we see that m = 4n + 3, n > 1. Moreover (R+ x M, dr2 + r2g) is hyper-Kahler, i.e., it has a quaternionic structure consisting of three global almost complex structures

218

13. 3-Sasakian Manifolds

which are Kahler with respect to the metric dr2 + r2 g. For a proof of the equivalence of these definitions see Boyer, Galicki, Mann [1994]. Again if M4n+3 has two Sasakian structures (0i, i, 77i, g), i = 1, 2 with j, and 2 orthogonal then the third structure defined by 3 = 0152 and 03X = -VX 3 gives a Sasakian 3-structure. Exploring this idea further, Tachibana and Yu [1970] prove the following theorem.

Theorem 13.2 Let M be a complete simply connected Riemannian manifold admitting Sasakian structures ((5i, iI 71i) g), i = 1, 2 such that g(1, S2) is a non-

constant function on M; then M is isometric to a unit sphere. The proof of this theorem is to differentiate the function f = g(j1, e2) twice and use the Sasakian conditions to show that f satisfies VlV kfj + 2figkj + fkglj + f jglk = 0 where f j = V j f . Then by a well-known theorem of Obata [1965] (see also Tanno [1978a]) M is isometric to a unit sphere. -ciX one readily obtains on a manifold with a 3-Sasakian Using Thus the subbundle spanned by {j1, 2, S3} is structure that [[i, j]. = integrable with totally geodesic leaves which are easily seen to be of constant etc., each leaf of the foliation curvature +1. Notice also that from is itself a 3-Sasakian manifold. The canonical example of a manifold with a Sasakian 3-structure is the sphere S4n+3 Its structure is readily obtained by taking S4,+3 as hypersurface in I.Eitn+l Each of the three almost complex structures forming the quaternionic structure of IH[n+1 applied to the outer normal of the sphere gives a vector field

i, i = 1, 2, 3 on S4n+3. These three vector fields are orthogonal and give rise to the standard Sasakian 3-structure on S1n+3 This Sasakian 3-structure on S4n+3 also projects under the Hopf fibration to the quaternionic structure on quaternionic projective space IHIPn. We will briefly discuss generalizations of this fibration below. We first however prove an early result of Kashiwada [1971] that a 3-Sasakian manifold is Einstein; this can be regarded as analogous to the well-known fact that a quaternionic Kahler manifold is Einstein.

Theorem 13.3 Let M4n+3 be a manifold with a Sasakian 3-structure i= 11273.

Then M4n+3 is an Einstein space with positive scalar curvature.

Proof. First taking an orthonormal basis {eA}, observe that from equation a) of Lemma 7.1, we have, summing on A, 1: g(RIAYZ, ceA) + A(Y, OZ) = - E P(eA, Y, Z, eA) = (dimM - 2)drl(Y, Z)

13.1 3-Sasakian manifolds

219

for any Sasakian structure (0, , ri, g). Now consider equation a) of Lemma 7.1 for the first Sasakian structure and set Z = eA and W = 02eA, i.e., g(RXYeA, cb14l2eA) + g(RXYcbleA, cb2eA) _ --P(X, Y, eA,

02P-

Now the idea is to sum on A, but in the second term assume that 1 is part of the basis and replace A)-In the first term replace 0102eA by 03eA +

the basis {eA} by {ql1eA, 1 }. The result of the sum is 2 E g(RxYeA, 03eA) =

4g(X, 03Y). Canceling the factor of 2 and applying the Bianchi identity we have g(ReAXY 03eA) +

g(ReAYX, 03eA) = 2g(X, ql3Y)

Using our first observation applied to the third Sasakian structure yields P(X, Cl3Y) = (4n + 2)g(X, 03Y)

which with the fact that Q = (dim M - 1)l; for any Sasakian structure gives the result. As a corollary, or from our previous remarks, we see that a 3-dimensional manifold with a Sasakian 3-structure is of constant curvature +1. Additional curvature properties of 3-Sasakian manifolds include the following. For a vector X orthogonal to { j1i 2, Z=3} (4n + 3 > 7), the sum of the three sectional c5-sectional curvatures satisfies 3

K(X, OjX) = 3, i=1

Tanno [1971]. If the sectional curvature of plane sections orthogonal to {Sl, S2, 3} is constant (4n + 3 > 7), the 3-Sasakian manifold is of constant curvature +1, Konishi and Funabashi [1976]. All complete 3-dimensional 3-Sasakian manifolds were classified by Sasaki [1972] in the following theorem.

Theorem 13.4 A complete 3-dimensional 3-Sasakian manifold is a quotient S3/I' where F is one of the following finite subgroups of Clifford translations on S3:

1. F = {I}, 2. F = {-4-I},

220 3.

13. 3-Sasakian Manifolds

I' is the cyclic group of order q > 2 generated by cos 2'r

L

1

- sin L' COs

2n

0

0

0

0

9

9

0

0

Cos

27r

0

0

sin

2n

- sin 2'q

4

cos 2"

q

q

4. P is a group of Clifford translations corresponding to a binary dihedral group or the binary polyhedral groups of the regular tetrahedron, octahedron or icosahedron. Boyer, Galicki and Mann [1994] classified all 3-Sasakian homogeneous spaces.

Theorem 13.5 Let M be a 3-Sasakian homogeneous space. Then M is one of the following: Sp(n + 1)

Sp(n + 1)

s4n+3

SP(n)

r..

RP4n+3

SP(n) X Z2

'

SU(m)

SO(k)

S(U(m - 2) x U(1))'

SO(k - 4) x Sp(1)'

G2

F4

E6

E7

E8

E7.

Sp(1)' Sp(3)' SU(6)' Spin(12)' For the first two cases when n = 0, Sp(0) is the identity group; m > 3; and k > 7. Moreover M fibres over a quaternionic Kahler manifold; the fibre is Sp(l) for S4n+3 and SO(3) in the other cases. There are in addition many inhomogeneous 3-Sasakian spaces, even simply connected ones. Boyer, Galicki and Mann [1994] prove that there are infinitely many homotopically distinct 7-dimensional strongly inhomogeneous compact simply connected 3-Sasakian manifolds. Let (M, ql, t=, g, g) be a Sasakian manifold with complete Riemannian metric

g and denote by I (M) the isometry group of g. Let A(M) be the automorphism group of the Sasakian structure (0, , rl, g) i.e., A(M) is the subgroup of isometries which also preserve 0, Z; and q. In [1970] Tanno proved the following theorem.

Theorem 13.6 Let (M, 0, , rl, g) be a complete Sasakian manifold which is not of constant curvature. Then either dim I (M) = dim A(M) or dim I (M) = dim A(M) + 2. Moreover dim I (M) = dim A(M) if and only if M does not admit a Sasakian 3-structure and dim I (M) = dim A(M) +2 if and only if M admits a Sasakian 3-structure.

13.1 3-Sasakian manifolds

221

Turning to fibration questions, Tanno [1971, p. 325] proves that of the three dimensional 3-Sasakian manifolds in Theorem 13.4, only S3(1) and RP3(1) are regular with respect to any of the three characteristic vector fields. Consequently we have the following result of Tanno [1971, p. 326].

Lemma 13.2 Let (M4n+s 02, , iti g), i = 1, 2, 3 be a complete Riemannian manifold with a K-contact 3-structure. If 1 is regular, then so are S2 and 3 and the leaves of the foliation induced by {S1, 2, 31 are isometric to S3(1) and RP3(1). Moreover M4n+3 is a fibre bundle over an almost quaternionic Kahler manifold M4n which is both Einstein and quaternionic Kahler since the structure is 3-Sasakian by Kashiwada's result (Theorem 13.1). These properties of 3Sasakian manifolds have been proven by various authors at different points in time. Under the assumptions of regularity, Tanno proved in [1971] that

the base manifold of the fibration is Einstein. When n > 2 and the structure is 3-Sasakian, Ishihara [1973] proved that the base manifold is quaternionic Kahler. Conversely if the induced structure on the base manifold is quaternionic Kahler, then the structure is 3-Sasakian, (Konishi [1973]). The

reason for the restriction n > 2 is that the usual definition of a quaternionic Kahler manifold M4n in terms of holonomy breaks down in dimension 4; more precisely, the condition that holonomy group be contained in Sp(n) Sp(l) (= Sp(n) x Sp(1)/{±I}) C SO(4n) becomes in dimension 4 simply the orientability of the manifold, since Sp(l) Sp(1) = SO(4). Thus in dimension 4 a manifold is said to be a quaternionic Kahler manifold if it is Einstein with non-zero scalar curvature and self-dual (see e.g., LeBrun [1991]).

With this understanding Tanno [1996] extended Ishihara's result to include n = 1. With Kashiwada's recent result in mind, we summarize this discussion in the following theorem.

Theorem 13.7 Let (M4n+3, i, g), i = 1, 2, 3 be a complete Riemannian manifold with a contact metric 3-structure. If one of the j1 's is regular, then M4n+3 is an Sp(l) or SO(3) bundle over a quaternionic Kdhler manifold

M4n.

The converse question was considered by Konishi in [1975], i.e., starting with

a quaternionic Kahler manifold M4n with n > 1, she constructs a canonical SO(3) bundle, M4n+3, over M4n. When the scalar curvature of M4n is positive, M4n+3 has a Sasakian 3-structure. Tanno [1996] extends this construction to

the case n = 1 and to the case of negative scalar curvature. We state these results as follows.

Theorem 13.8 Let M4n be a quaternionic Kahler manifold with non-zero scalar curvature. Then there exists a canonical SO(3) bundle, M4"+3, over M4n. If the scalar curvature of M4n is positive, M4n+3 admits a Sasakian 3-

222

13. 3-Sasakian Manifolds

structure; if the scalar curvature is negative M4n+3 admits a pseudo-Sasakian 3-structure, the signature of the metric g being (3, 4n). In dimensions 7 and 11 the complete principal Riemannian fibrations with Sasakian 3-structure of positive scalar curvature were determined by Boyer,

Galicki and Mann [1993, p.253]. They are: S7, RP7 and SU(3)/U(i) in dimension 7, and Sll RP", SU(4)/S(U(2) x U(1)) and G2/SU(2) in dimension 11.

In general the foliation determined by the vector fields 1S1, 2, 31 is not a fibration and Boyer, Galicki and Mann [1994] prove the following result. i = 1 2 3 be a 3-Sasakian manifold such that the vector fields S1, S2, S3 are complete; then the space of leaves of

Theorem 13.9 Let M4n+3

the foliation determined by f 1, e2, 3} is a quaternionic Kahler orbifold of dimension 4n with positive scalar curvature 16n(n + 2). In view of our study of twistor spaces as complex contact manifolds over quaternionic Kahler manifolds, we also consider fibrations of a manifold with a contact metric 3-structure by one of the structure vector fields. The following result of Ishihara and Konishi [1979] is not surprising.

Theorem 13.10 Let M4n+3 be a manifold with a contact metric 3-structure and suppose that one of the structures, say (01, S1) 771, g), is regular and Sasakian; then the orbit space M'+3/e1 admits a complex contact metric structure. Moreover if M4n+3 is a 3-Sasakian manifold, the complex contact metric

structure on the orbit space is a Kiihler-Einstein structure of positive scalar curvature.

With Kashiwada's result the last statement is true without the additional hypothesis. Without the regularity, this question was taken up by Boyer and Galicki [1997] who gave an orbifold version and showed that the space Z = M4n+3/j is the twistor space of a quaternionic Kahler orbifold. Some important results on the topology of a compact 3-Sasakian manifold are the following due to Galicki and Salamon [1996]. In Section 6.8 we noted that the odd Betti numbers of a compact Sasakian manifold are even up to middle dimension. Galicki and Salamon prove a stronger result for 3-Sasakian manifolds.

Theorem 13.11 Let M4n+3 be a compact 3-Sasakian manifold. Then the odd Betti numbers b2k+1 are all zero for 0 < k < n. In the regular case Galicki and Salamon [1996] relate the Betti numbers of M4n+3 to those of the quaternionic Kahler base space M4n and the intermediate space Z obtained as the Boothby-Wang fibration with respect to one of the characteristic vector fields. In particular the odd Betti numbers of these

13.2 Integral submanifolds

223

spaces vanish (cf. Salamon [1982]). The complex structure on Z leads to the fact that Rp4n+3 is the only compact regular 3-Sasakian manifold which is not simply-connected. Galicki and Salamon also prove the following remarkable result. Theorem 13.12 Up to isometries, there are only finitely many compact regular 3-Sasakian manifolds in each dimension 4n+3, n > 0 and the only compact regular 3-Sasakian manifolds with b2 > 0 are the spaces U(m)

U(m - 2) x U(1)'

m>3.

Galicki and Salomon also obtain results on certain sums of Betti numbers and the relation b2k(M4n+3) = b2k(Z) - b2k-2(Z) = - b2k-4(M4"), k < n. Again an excellent survey of recent work on 3-Sasakian manifolds is Boyer and b2k(M4n.)

Galicki [1999].

13.2

Integral submanifolds

When we have one contact structure on a manifold of dimension 2n + 1 we have seen that the maximum dimension of an integral submanifold is n. In the present context we have three independent contact structures on a manifold of dimension 4n + 3 and we begin with the following lemma. Lemma 13.3 The maximum dimension of a submanifold which is an integral submanifold of all three contact structures is n.

Proof. If X1,. .. Xr is a local basis tangent to such a submanifold, then ,

the cb2Xa, i = 1, 2, 3, a = 1, . , r are normal to the submanifold as well as perpendicular to j, i = 1, 2, 3 since Oj j = Sk. Moreover from c5k = Oz0i-71j these vectors are independent. Thus the codimension is at least 3r + 3 and .

.

hence r < n. Thus in dimension 7 these would be integral (Legendre) curves and that they are plentiful can be seen as follows. The characteristic vector fields of the standard Sasakian 3-structure on S7 are tangent to the fibres S3 of S7 viewed as a principal S3-bundle over S4. Thus the horizontal lift of a curve on S4 gives a curve in S7 which is a Legendre curve for each of the three contact structures. Curves in S7 which are Legendre curves for all three contact structures and of constant curvature and unit torsion were classified by Baikoussis and Blair in [1995].

In contrast to this lemma it is possible to have a submanifolds of dimension up to 2n + 1 which are integral submanifolds of two of the three contact

structures. We will discuss this briefly in the case of S7. Note first that if

224

13. 3-Sasakian Manifolds

M3 is a 3-dimensional submanifold of a manifold with Sasakian 3-structure which is an integral submanifold of the first Sasakian structure and an invariant submanifold with respect to the third Sasakian structure (03, 6, 773, g), then it is an integral submanifold of the second Sasakian structure (02, 62, 772, g)(. To see this let X be tangent to M3; then 772(X) = g(e2, X)

g(ql36, X) = -g(j1, q3X) = -771(03X) = 0. To give a couple of examples consider Euclidean space E8 with three complex structures (or quaternionic structure),

I= ( 14

I,

04

J=

0

0

0

12

0

0

-12

0

0

12

0

0

-12

0

0

0

,

K = -IJ

where I, denotes the n x n identity matrix. (The sign conventions of Baikoussis and Blair [1995] differ from the above.) Let x denote the position vector of the

unit sphere in E8 and as usual define three vector fields on S7 by ji = -Ix, S2 = -Jx, l;3 = -Kx. The dual 1-forms 77z are three independent contact structures on S7. Now consider the linear subspace L of E8 defined by x5 = xs = x7 = x8 = 0. The intersection S3 = S7nL is then an integral submanifold for 77, and 772, but 3 is tangent and this sphere is invariant under the action of K restricted to S. Now consider the torus in the 3-sphere we just constructed defined by x1 + x3 = x2 + x4 = This torus is an integral surface for both 77i and 772 but l;3 is tangent. Note also that the image of the tangent space under 03 is normal. We now state the following result from Baikoussis and Blair [1995]. 2,

2.

Theorem 13.13 Let M3 be a 3-dimensional submanifold of S7 isometrically immersed as an integral submanifold of the Sasakian structure S1, 71i, 9) (and (02, 6, 772, 9)) and an invariant submanifold of the third Sasakian structure (03, S3, 773, 9). Then M3 is a principal circle bundle over a holomorphic Legendre curve in CP3. Moreover if M3 is of constant cl-sectional curvature, then either M3 is totally geodesic or a principal circle bundle over the holomorphic (Calabi) curve CP1(3) As a corollary we note that a holomorphic Legendre curve of constant Gaus-

sian curvature in CP3, cannot lie in a totally geodesic subspace CP2. Similarly if M3 is a 3-dimensional submanifold of S7, isometrically immersed as an integral submanifold of one of the Sasakian structures and as an invariant submanifold of one of the other Sasakian structures, and if M3 is of constant 0sectional curvature and not totally geodesic, then M3 cannot lie in any totally geodesic Sasakian submanifold (SS) of S7. We now turn to the case of a 2-dimensional submanifold of S7 which is an integral submanifold of two of the Sasakian structures, Even though it cannot

13.2 Integral submanifolds

225

be an integral submanifold for the third structure, we still can have that 03 maps the tangent bundle into the normal bundle as in the example of a torus. In this regard we have the following result of Baikoussis and Blair [1995].

Theorem 13.14 Let M2 be a surface isometrically immersed in S7 as an integral submanifold of the Sasakian structures (01, 1, 771, 9) and (02,

2, 712, 9)

and suppose that 03(TM) C T1M. Then M2 is flat and 3 is tangent to M2.

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Subject Index

(n, µ)-nullity distribution, 102 (0, , ??)-structure, 33 *-Ricci form, 164 *-Ricci tensor, 97, 162 *-scalar curvature, 97, 162

GH- sectional curvature, 204 D-homothetic deformation, 103, 114 ,q-Einstein, 105 n-nullity distribution, 105 0-basis, 34 0-geodesic, 115 1>-geodesic symmetry, 115

1-sectional curvature, 110 'H-homothetic deformation, 205 "total scalar curvature", 163, 208 c-sectional curvature, 177 3--r-manifold, 183 3-Sasakian manifold, 216

Abbena metric, 5, 174 almost complex structure, 31 almost contact 3-structure, 215

almost contact metric 3-structure, 216

almost contact metric structure, 33 almost contact structure, 33 almost coquaternion structure, 215 almost Hermitian structure, 3, 31 almost Kahler structure, 3, 31 almost regular, 29

Anosov directions, 180 Anosov flow, 179 Anosov-like, 183 anti-invariant submanifold, 62 associated metric on a contact manifold, 36

associated metric on a symplectic manifold, 3, 36 associated metric, complex contact, 191

Boothby-Wang fibration, 26 Brieskorn manifold, 85 C-parallel, 131 C-totally real submanifold, 55, 128 Calabi imbeddings, 124, 212 characteristic class of S1-bundle, 16 characteristic subbundle, 191 characteristic vector field, 17 Codazzi equation, 121 coeffective p-form, 89 coisotropic, 7 compatible metric, 33 complex almost contact metric manifold, 191

complex contact manifold, 190 complex contact metric manifold, 192 complex contact space form, 204 complex contact subbundle, 190 complex Sasakian manifold, 204

254

Subject Index

complex symplectic manifold, 198 conformally Anosov flow, 184 connection form, 15 connection map, 138

connection on a principal bundle, 14

contact circle, 25 contact distribution, 17, 55 contact manifold, 17 contact metric 3-structure, 216 contact metric structure, 36 contact Riemannian structure, 36 contact structure in the wider sense,

Hermitian metric, 3, 31 Hermitian Ricci tensor, 161 holomorphic Frenet curve, 209 holomorphic integral submanifold, 192

holomorphic Legendre curve, 192 holomorphically planar curve, 83 homogeneous contact metric manifolds, 118 horizontal, 14 horizontal lift, 15, 137, 149 horizontal subbundle, 190 hyper-Kahler manifold, 37

18

contact subbundle, 17, 55 contact transformation, 18, 57 contact type, 22 contact Whitney sphere, 61 cosymplectic, 77 CR-manifold, 72, 76 CR-submanifold, 73

infinitesimal contact transformation, 57

integral submanifold, 55, 192 invariant submanifold, 122 isotropic, 6 Iwasawa manifold, 194 J-basis, 32

Douady portrait, 24 effective p-form, 88

first normal space, 210 Frenet curve, 131 fundamental 2-form, 3, 31

gauge transformation, 173 Gauss equation, 121 generalized Tanaka connection, 170 generalized Tanaka-Webster scalar curvature, 171 generating function, 5

K-contact structure, 52, 67, 70 Kenmotsu manifold, 80 Kodaira-Thurston manifold, 4 Kahler structure, 3 Lagrangian H-umbilical submanifold, 127

geodesic flow, 143 Goldberg conjecture, 162

Lagrangian submanifold, 5 Legendre curve, 18, 60 Levi form, 72 Liouville form, 2 locally D-symmetric space, 115 locally Killing-transversally symmetric space, 120

half-conformally flat, 196 Hamiltonian vector field, 3 Heisenberg group, 49 Hermitian manifold, 31

nearly cosymplectic structure, 84 nearly Kahler structure, 51, 83 nearly Sasakian structure, 85 normal almost contact structure, 64

Subject Index

normal complex contact metric structure, 203 normalized contact structure, 191 overtwisted, 25

periodic point, 28 principal G-bundle, 11 principal circle bundle, 11 projectively Anosov flow, 184 pseudo-Hermitian manifold, 72, 76

quasi-Kahler structure, 3 quaternionic Kahler manifold, 196, 221

R-contact manifold, 29 Reeb vector field, 17 regular contact structure, 18 Ricci-Kiihn, equation of, 121 Salamon-Berard-Bergery metric, 197 Sasaki metric, 138, 149 Sasakian 3-structure, 216 Sasakian globally q-symmetric space, 116

Sasakian locally q-symmetric space, 115

Sasakian manifold, 69, 79 Sasakian space form, 113 second normal space, 210

255

semi-Kahler structure, 208 special directions, 178 strict complex contact manifold, 190 strict contact transformation, 18, 57 strict infinitesimal contact transformation, 57 strongly locally 0-symmetric space, 117

strongly pseudo-convex CR-manifold, 72, 76 symplectic manifold, 1 symplectic vector field, 9 symplectomorphism, 1

taut contact circle, 25 Thurston manifold, 4, 174 tight contact structure, 25 totally real submanifold, 6, 124 trans-Sasakian structure, 81 tridegree, 89 twistor space, 196 Veronese imbeddings, 124, 212 vertical lift, 137, 149 vertical subbundle, 191

weakly locally q5-symmetric space, 117

Webster metric, 72 Webster scalar curvature, 118, 171, 172

Author Index

Abbena, E., 4, 174 Abe, K., 86, 87 Alekseevskii, D. V., 196 Altschuler, S., 23 Anosov, D. V., 179, 180, 183 Apostolov, V., 45, 162 Armstrong, J., 162 Auslander, L., 179

Baikoussis, C., 131-133, 135, 209, 211,215,223-225 Bang, K., 107, 108, 141, 148, 151, 153

Banyaga, A., 29 Bejancu, A., 73 Belgun, F. A., 72 Bennequin, D., 24 Berger, M., 46, 157, 196 Berndt, J., 119 Besse, A. L., 197 Bishop, R. L., 11, 15

Blair, D. E. (=author), 39, 43, 44, 62, 73, 77, 80, 83-85, 87, 88,

92, 99-103, 105, 107, 108, 115, 117, 119, 120, 131-133,

135,142,145-147,160-166, 168, 172, 177, 180, 183-187,

192,209,211-213,215,223225

Boeckx, E., 102, 104, 105, 117, 118

Boothby, W. M., 10, 26, 59, 142, 189, 190

Borisenko, A. A., 142, 150 Borrelli, V., 61, 127 Bouche, T., 89 Boyer, C. P., 79, 106, 215, 217, 218, 220, 222, 223 Brieskorn, E., 86 Bryant, R. L., 211 Bueken, P., 117, 118, 120

Calabi, E., 79, 210, 212 Calvaruso, G., 69, 108, 118 Capursi, M., 80 Carriazo, A., 62 Cartan, E., 146, 169 Chen, B.-Y., 6, 61, 73, 117, 125, 127, 129

Chen, H., 105 Chern, S. S., 33, 76, 123, 166, 171, 209

Chevalley, C., 35 Chinea, D., 77, 88, 89, 122 Cho, J., 119 Chuman, G., 157 Cordero, L. A., 5 Cowen, M. J., 209

Crittenden, R. J., 11, 15 Davidov, J., 162 de Leon, M., 5, 77, 88, 89

258

Author Index

Deng, S., 169 Dillen, F., 131, 212 do Carmo, M. P., 123 Dolbeault, S., 209 Dombrowski, P., 71, 138 Douady, A., 24 Draghici, T., 45, 162-164 Dragomir, S., 72

Ebin, D., 38, 46, 169 Eckmann, B., 79 Ejiri, N., 125 Eliashberg, Y., 25, 184, 185 Endo, H., 122, 123 Erbacher, J., 87 Ferndndez, M., 5, 89, 194 Foreman, B., 190-192, 197-200, 203, 204, 206, 207 Fujita, H., 120 Fujitani, T., 87, 89 Funabashi, S., 219 Galicki, K., 79, 106, 215, 217, 218, 220, 222, 223 Geiges, H., 23, 25, 67, 69, 72 Ghosh, A., 108 Ghys, E., 180 Ginzburg, V., 28 Goldberg, S. I., 63, 84, 87, 88, 162, 190

Gompf, R., 5 Gonzalo, J., 20, 25, 69, 186 Gonzalez-Ddvila, J. C., 120 Gonzalez-Ddvila, M. C., 120 Gouli-Andreou, F., 108, 183, 209, 211

Gray, A., 5, 31, 81, 163, 194 Gray, J., 18, 19, 21, 33 Green, L., 179 Greenfield, S., 72 Gromov, M., 7

Hadjar, A., 23 Haefliger, A., 3 Hahn, F., 179 Hamilton, R. S., 76, 166, 171 Harada, M., 123, 124 Hasegawa, I., 99 Hatakeyama, Y., 10, 34, 58, 64, 65, 82, 92, 93, 142, 192 Hervella, L. M., 31, 81, 163 Hilbert, D., 157 Hitchin, N. J., 216 Hofer, H., 28 Holubowicz, R., 5 Houh, C.-S., 125, 203 Hsu, C. J., 86, 138 Huygens, C., 19 Ianus, S., 74, 161, 162 Ibanez, R., 89 Ikawa, T., 129 Ishihara, S., 189, 191, 192, 195, 196, 202, 216, 221, 222 Itoh, M., 53

Janssens, D., 81 Jayne, N., 193, 194 Jelonek, W., 5 Jerison, D., 173 Jimenez, J. A., 119 Kashiwada, T., 216, 218 Kenmotsu, K., 80, 123 Kobayashi, S., 11, 15, 16, 27, 31, 32, 63, 123, 189, 190, 192

Kodaira, K., 4, 63 Kon, M., 123, 124, 129, 153 Konishi, M., 189, 191, 192, 195, 202, 216, 219, 221, 222

Korkmaz, B., 200, 202-205, 213 Koufogiorgos, Th., 102 Koufogiorgos, Th.,103, 105, 107, 108, 113, 120, 132, 133, 147, 148, 172, 184

Author Index

Kowalski, 0., 119, 120, 139, 140 Kriegl, A., 10, 56, 59 Kuo, Y.-Y., 215, 216 Lawson, H. B., 210 LeBrun, C., 189, 190, 196, 198, 221 Ledger, A. J., 160, 164, 165 Lee, J. M., 173 Libermann, P., 7-9, 77 Lie, S., 19 Ludden, G. D., 125, 192 Lutz, R., 23, 87 Lychagin, V., 56 MacLane, S., 19 Mann, B. M., 79, 218, 220, 222 Marie, C.-M., 7-9 Marrero, J. C., 77, 81, 88, 89

Martinet, J., 7, 20 McCarthy, J. D., 5 McDuff, D., 5 Meckert, C., 87 Michor, P. W., 10, 56, 59 Milnor, J., 118, 181 Mitric, G., 147 Mitsumatsu, Y., 184, 185 Miyahara, Y., 129 Monna, G., 19 Morimoto, A., 79, 82 Morimoto, S., 106 Moroianu, A., 162, 189 Morrow, J., 63 Morvan, J.-M., 7, 61, 127 Moser, J., 7 Moskal, E., 87-89, 111 Mozgawa, W., 5 Muskarov, 0., 162 Musso, E., 147 Muto, Y., 157, 159, 160 Myers, S. B., 99

Nagano, T., 157

259

Nakagawa, H., 212 Newlander, A., 63 Nirenberg, L., 63 Nomizu, K., 11, 15, 31, 63 Nurowski, P., 162

O'Neill, B., 115 Obata, M., 218 Ogawa, Y., 87, 92, 93 Ogiue, K., 6, 61, 112, 115, 123-125, 129, 212

Oh, W. T., 174 Okumura, M., 50, 71, 84, 85, 106, 108, 114, 116, 125, 139 Olszak, Z., 95, 97, 98 Oubina, J. A., 78, 80, 81

Padron, E., 77 Palais, R. S., 18, 26 Papantoniou, B. J., 102, 103, 105 Park, J.-S., 174 Pastore, A. M., 107, 145 Perrone, D., 69, 88, 107, 108, 118120, 146, 147, 163, 165, 166,

170-172, 183, 185-187 Petkov, V., 28 Popov, G., 28 Priest, E. R., 110 Przanowski, M., 162

Rabinowitz, P. H., 28 Reckziegel, H., 150 Reeb, G., 17, 22 Ros, A., 61, 124, 127 Rukimbira, P., 29, 94, 99, 100, 120, 169

Salamon, S., 189, 197, 222, 223 Salingaros, N. A., 110 Sasaki, S., 19, 22, 55, 56, 64, 65, 86, 87, 138, 142, 173, 219

Sato, H., 67, 87 Seino, M., 99

260

Author Index

Sekigawa, K., 106, 162 Semmelmann, U., 189 Sharma, R., 100, 105, 107, 108, 120, 184

Shibuya, Y., 115, 119, 192 Showers, D. K., 85 Sikorav J.-C., 7 Simons, J., 123 Smolentsev, N. K., 46 Steenrod, N., 60 Stong, R. E., 19, 20 Tachibana, S., 71, 87, 139, 216, 218 Takahashi, T., 86, 115, 116 Tanaka, N., 170 Tanno, S., 27, 52, 74, 82, 87, 88, 92, 93, 103, 105-108, 114, 120, 143, 145, 147, 148, 167, 170, 171, 173, 174, 218-221 Tashiro, Y., 49, 78, 85, 142, 144 Thomas, C. B., 29, 169 Thurston, W., 4, 25, 184, 185 Tischler, D., 6 Tricerri, F., 147 Tsichlias, C., 105

Udriste, C., 215 Urbano, F., 61, 124, 125, 127 Vaisman, I., 86 Van de Ven, A., 86 Vanhecke, L., 81, 108, 115, 117-120 VanLindt, D., 124, 131

Verheyen, P., 124, 131 Vernon, M., 85 Verstraelen, L., 124, 131, 212 Viterbo, C., 28

Vitter III, A. L., 209 Vrancken, L., 131, 212, 213

Walters, P., 179 Wang, H. C., 26, 142 Watanabe, Y., 119, 120 Watson, B., 5 Wegrzynowski, S., 119 Weinstein, A., 6, 7, 28, 61 Weyl, H., 2 Wolf, J. A., 22, 189 Wolfson, J. G., 5 Wood, C. M., 175 Woodhouse, N., 7 Wu, H., 31, 32, 63 Wu, L., 23

Xenos, Ph. J., 108, 183 Yamaguchi, S., 129, 157 Yampol'skii, A. L., 142, 143, 150 Yano, K., 31, 63, 85, 125, 153, 216 Yau, S.-T., 6, 124 Ye, Y.-G., 189, 190 Yu, W. N., 216, 218 Zeghib, A., 99 Zehnder, E., 28

l)as id E. Blair

Rientannian (eometry of Contact and SN rnplectic Manifolds

This monograph deals ssith the Riemannian geontelr% of both s) mplectic and contact manifolds. ssith particular emphasis on the latter. The text is carefull% presented. Topics unfold systematically from Chapter 1. sshich examines the general theory ofs%ntplectic manifolds. Principal circle

bundles (Chapter 2) are then discussed as a prelude to the Boothhy-Tang libration of a compact regular contact manifold in Chapter 3. sshich deals ss ith the general theory of contact manifolds. Chapter 4 focuses on the general setting of Rienusnnian metrics associated ssith both sy mplectic and contact structures, and Chapter 5 is denoted to integral submanitolds of the contact subhundle. Topics treated in the subsequent chapters include Sasakian manifolds, the important study of the curvature of contact metric manifolds. submanifold theor% in both the K:ihler and Sasakian settings, tangent sphere bundles, curvature functionals. complex contact manifolds and 3-Sasakian manifolds. The book serves both as a general reference for mathematicians to the basic properties ofs%mplectic and contact manifolds and as an excellent resource for graduate students and researchers in the Riemannian geometric arena. The prerequisite for this text is a basic course in Riemannian geometry.

Birkhiiuser ISBN II-8176-4261-7 syvyss.hirkhauser.com