Locally Conformal Kahler Geometry (Progress in mathematics)

Progress in Mathematics

Sorin Dragomir Liviu Ornea

Locally Conformal Kahler Geometry

Birkhauser

Progress in Mathematics Volume 155

Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

Sorin Dragomir Liviu Ornea

Locally Conformal Geometry

Birkhauser Boston Basel Berlin

Sorin Dragomir Dipartimento di Matematica University degli Studi della Basilicata 85100 Potenza, Italia

Liviu Ornea Facultatea de Matmatica Universitatea din Bucuresti Bucure,§ti, Romania

Library of Congress Cataloging-in-Publication Data Dragomir, Sorin, 1955Locally conformal Khhler geometry / Sorin Dragomir, Liviu Omea. cm. -- (Progress in mathematics ; v. 155) p. Includes bibliographical references. ISBN 0-8176-4020-7 -- ISBN 3-7643-4020-7 1. Khhlerian manifolds. 2. Geometry, Differential. I. Omea, II. Title. III. Series: Progress in mathematics Liviu, 1960. (Boston, Mass.) ; vol. 155 97-27397 QA649.D76 1997 CIP 515'.73--dc2l

AMS Classification Codes: 53D20, 53C15, 53C40, 53C56.

Printed on acid-free paper ® 1998 Birkhauser

Birkhl user

Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhduser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhguser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-4020-7 ISBN 3-7643-4020-7 Reformatted from authors' disks by TsXniques, Inc. Boston, MA Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the United States of America

987654321

Contents Introduction 1

ix

L.c.K. Manifolds

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2 Principally Important Properties 2.1 2.2 2.3

Vaisman's conjectures . Reducible manifolds Curvature properties . Blow-up . . . . . . . . . An adapted cohomology

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3 Examples 3.1 Hopf manifolds .. ... ..

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3.3 A generalization of Thurston's manifold 3.4 A four-dimensional solvmanifold . . . . 3.5 SU(2) x S1 . .. 3.6 Noncompact examples . . . . . . . . . . 3.7 Brieskorn & Van de Ven's manifolds . .

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4 Generalized Hopf manifolds

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5 Distributions on a g.H. manifold

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6 Structure theorems

.. L.c.K.0 manifolds . ......... ............ . 6.3 A spectral characterization .. .. 6.4 k-Vaisman manifolds .. 6.1 6.2

Regular Vaisman manifolds

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49 56 60 66

CONTENTS

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7 Harmonic and holomorphic forms 7.1 7.2

Harmonic forms . . . . . . Holomorphic vector fields

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8 Hermitian surfaces

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9 Holomorphic maps

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9.1 9.2 9.3

General properties . . Pseudoharmonic maps A Schwarz lemma . . .

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10 L.c.K. submersions

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107 111

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10.1 Submersions from CH' 10.2 L.c.K. submersions . . . . . . . . . . . . . . . . . . . . . . 10.2.1 An almost Hermitian submersion with total space Stn-1(c, k) 10.2.2 An almost Hermitian submersion with total space

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x R, k > -3c2 ................ 125

R211-1(c)

x R ....................... 126

10.2.3 An almost Hermitian submersion with total space

(R x Bn-1)(c,k) x R, k < -3c2 10.3 Compact total space . . . . 10.4 Total space a g.H. manifold

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11 L.c. hyperKahler manifolds

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12 Submanifolds 12.1 Fundamental tensors

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12.2 Complex and CR submanifolds ................. 153 12.3 Anti-invariant submanifolds . . 12.4 Examples .

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12.6 Totally umbilical submanifolds ................. 172 13 Extrinsic spheres 13.1 Curvature-invariant submanifolds . 13.2 Extrinsic and standard spheres . .

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vii

CONTENTS

14 Real hypersurfaces

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. 14.1 Principal curvatures . 14.2 Quasi-Einstein hypersurfaces 14.3 Homogeneous hypersurfaces . 14.4 Type numbers .. . 14.5 L. c. cosymplectic metrics .

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15.3 L.c.K. and Kahler submanifolds ................. 251 15.5 Planar geodesic immersions

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16 Integral formulae

.. . ... ... .. .. 16.1 Hopf fibrations 16.2 The horizontal lifting technique .... ..... .. ...... .. .. .. .... . . ..... 16.3 The main result . .

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257 257 260 267

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17 Miscellanea 17.1 Parallel Ilnd fundamental form .

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17.8 Geodesic symmetries . 17.9 Submersed CR submanifolds

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A Boothby-Wang fibrations

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B Riemannian submersions

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Bibliography

307

To the memory of Franco Tricerri

Introduction Let (M2,, J, g) be a Hermitian manifold of complex dimension n, where J denotes its complex structure, and g its Hermitian metric. Cf. P. Libermann, [174], [175], g is a locally conformal Kdhler (l.c.K.) metric if g is conformal to some local Kahlerian metric in the neighborhood of each point of M2n. Precisely, g is l.c.K. if there is an open cover {Ut}jEI of Mgr and a family {ft}AEI of C°O functions f; : U1 --, R so that each local metric

exp(-ff )gw;

is Kahlerian. The main theme of this book is the study of l.c.K. manifolds, i.e. manifolds which carry some U.K. metric. Nowadays complex geometry deals primarily with Kahlerian manifolds, cf. e.g. [291], [21], i.e. manifolds carrying some Kahlerian metric. Nevertheless, some readily available complex manifolds, such as complex Hopf manifolds (cf. [132]) admit no global Kahlerian metrics at all. Indeed, let

A E C, 0 < IAI < 1, and n E Z, n > 2. Let Da be the 0-dimensional Lie group generated by the transformation z --+ Az, z E Cr - {0}. Then (cf. e.g. [162], vol. II, p. 137) Da acts freely on Cr - {0} as a properly discontinuous group of complex analytic transformations of Cr - {0}, so that the quotient space CHa = (Cr - {0})/Aa becomes a complex manifold. This is the complex Hopf manifold. Since CHa __ S1 X

S2n-1 (a diffeomorphism)

CHa is compact and its first Betti number is bl (CHa) = 1. Then (in view of Theorem 5.6.2 in [106], p. 178) CHa admits no globally defined Kahlerian metrics. In turn (as discovered by W.M. Boothby, [36], for n = 2) any complex Hopf manifold CHa admits a globally defined l.c.K. metric go. Indeed, the Hermitian metric Izi_2b6kdzi

® dzk

x

INTRODUCTION

on C" - {0} is Da-invariant, and therefore it induces a global Hermitian metric go on CH,;, the Boothby metric. As observed later by I. Vaisman (cf. [2691) go turns out to be 1.c.K. Several examples of complex manifolds

admitting no global Kahlerian metrics (yet carrying natural global l.c.K. metrics) were subsequently discovered, cf. e.g. the complex Inoue surfaces SM with the Tricerri metric, cf. (258]. The first eleven chapters of this book report on the main achievements in the theory of l.c.K. manifolds. The last six chapters present the theory of submanifolds in 1.c.K. manifolds, as developed by J.L. Cabrerizo & M.F. Andres, [46], S. Ianu§ & K. Matsumoto & L. Ornea, [133], [134], K. Matsumoto, [184], [185], F. Narita, [198], and the authors, cf. [216], [217], and [75], (76], [77], for whom the geometry of the second fundamental form of a submanifold (in particular, of a CR submanifold, in the sense of A. Bejancu, [15]) has been the main interest for quite a few years. The geometry of l.c.K. manifolds has developed mainly since the 1970s, although, as indicated above, there are early contributions by P. Libermann

(going back to 1954). The recent treatment of the subject was initiated by I. Vaisman in 1976. In a long series of papers (cf. [266], [268]-[278]), he established the main properties of 1.c.K. manifolds, demonstrated a connection with P. Gauduchon's standard metrics, and recognized the Boothby metric as l.c.K., thus being naturally led to the introduction of the generalized Hopf (g.H.) manifolds. He also explained the relationship between this class of 1.c.K. manifolds and the contact metric manifolds. Next, F. Tricerri proved that the blow-up at a point preserves the class, and together with I. Vaisman (cf. [260]) studied the curvature properties of l.c.K. surfaces. A great amount of research has been produced by E. Bedford & T. Suwa, [20], A. Cordero & M. Fernandez & M. De Leon, [68], T. Kashiwada, [147][151], T. Kashiwada & S. Sato, [153] (who showed that the first Betti number of a compact g.H. manifold is odd), B.Y. Chen & P. Piccinni, [64] (who studied foliations naturally occuring on a l.c.K. manifold), S.I. Goldberg & I. Vaisman, [115], C.P. Boyer, [40] (who demonstrated the relationship between anti-self-dual compact complex surfaces and l.c.K. surfaces), H. Pedersen & Y.S. Poon & A. Swann, [226] (who proved that Hermite-Einstein-Weyl manifolds are g.H. manifolds), M. Pontecorvo, [231] (who studied conformally flat l.c.K. surfaces), D. Perrone, [227] (who gave a spectral characterization of complex Hopf surfaces), K. Tsukada, [262] (who studied holomorphic vector fields on g.H. manifolds), J.C. Marrero & J. Rocha, [179] (who studied submersions from a 1.c.K. manifold), etc.

The most important of the authors' contributions seem to be the (partial) classification of totally umbilical submanifolds (in particular of extrinsic

INTRODUCTION

xi

spheres) in a g.H. manifold (cf. [134] and [881), and the classification of all compact minimal CR submanifolds, generically embedded in a complex Hopf manifold with the Boothby metric, which are fibered in tori, and have a flat normal connection, and a second fundamental form of constant length (cf. 111D-

The second named author has had useful discussions with P. Gauduchon, S. Ianu§, L. Lemaire, S. Marchiafava, S. Papadima, P. Piccinni, and V. Vuletescu. While this book was written, both authors profited from many useful comments from I. Vaisman. The authors wish to express their gratitude to all these people. The interest of the first named author in 1.c.K. geometry has been stimulated by discussions with F. Tricerri. Both authors wish to dedicate this book, as a modest reminder, to his memoryl.

Potenza, 22 January 1997

1F. Tricerri died together with his wife and two children in an airplane crash in June 1994.

Locally Conformal Kahler Geometry

Chapter 1

L.c.K. Manifolds In this chapter we state several equivalent definitions of the notion of a locally conformal Kahler manifold and study the elementary emerging properties. Let (M2n, J, g) be a complex n-dimensional Hermitian manifold, where J denotes its complex structure and g its Hermitian metric. Then (M2n, J, g) is a locally conformal Kahler (1.c.K.) manifold if there is an open cover {Ui}iEt of M21 and a family {fi}jEJ of CI functions fi : Ui R so that each local metric

gi = exp(-fi)glu

(1.1)

is Kahlerian. Here gnu, = t! g where ti : Ui - M2n is the inclusion. Also (M2n, j, g) is globally conformal Kahler (g.c.K.) if there is a C°O function

f : M2n -p R so that the metric exp(f)g is Kahlerian. Let 1, Qi be the 2-forms associated with (J, g) , (J, g;) respectively (i.e. Sl(X, Y) = g(X, JY) for any X,Y E T(M2n), etc.). Then (1.1) yields

Qi = exp(-fi)HIu:.

(1.2)

Theorem 1.1 The Hermitian manifold (M2n, j, g) is 1.c.K. if and only if there exists a globally defined closed 1-form w on M2n so that

d1=wASl. Proof. Let us take the exterior differential of (1.2) to yield as df2i = 0

dil=df2AS1

on U. Thus

(dfi - dfj)Af = 0

(1.3)

CHAPTER 1. L.C.K. MANIFOLDS

2

on U1j = U; fl U. Therefore (as 0 is nondegenerate) df; = df; on U1, so that the local 1-forms df; glue up to a (globally defined) closed 1-form w on M2n so that w1u, = df;. Conversely, let w be a closed 1-form on M2n satisfying (1.3). By the classical Poincare lemma there is an open cover {U:}ZEI of M2n and a family of C' functions f= : U; R so that w = df; on U1. By (1.3) one has di l = df; A fl on Ut so that (by multiplying both members with

exp(- ft)) d(exp(- f;)S2) = 0, that is, exp(- f;)g is a Kiihler metric on U;, Q.E.D.

The closed 1-form furnished by Theorem 1.1 is the Lee form of the l.c.K. manifold M2n. Cf. H.C. Lee, [169], for the choice of terminology. By (1.2) any l.c.K. manifold is in particular a locally conformal symplectic manifold, e.g. I. Vaisman, [278]. Also (M2n, j, g) is globally conformal Miler (respectively Kahler) if the Lee form w is exact (respectively if w = 0). Thus any simply connected l.c.K. manifold is g.c.K. and the universal covering space of a 1.c.K. manifold is g.c.K.

Let (M2n, J, g) be a Hermitian manifold (n > 1). Let 6 = d` be the formal adjoint of d (the exterior differentiation operator) with respect to g and define a 1-form w on M2n by setting

w=n1(bQ)oJ. If a globally defined 1-form w satisfying (1.3) exists then it is uniquely deter-

mined and it is expressed by (1.4). Also, if M4 is a Hermitian surface (i.e. n = 2) then the 1-form (1.4) satisfies (1.3) yet generally it is not closed, cf. 1. Vaisman, [276].

The 1-form (1 - n)w is referred to as the torsion 1 -form in [103] (due to the fact that (812) o J turns out to be the trace of the torsion of the Chem connection of (M2n, j, g)). Let (M2n, J, g) be a l.c.K. manifold. Let B = wl -be the Lee vector field, where d denotes the raising of indices with respect to g (i.e. g(X, Aa) = \(X) for any X E T(M2n) and any 1-form A on M2n). Let V be the Levi-Civita connection of (M2n, g).

Theorem 1.2 The Levi-Civita connections D' of the local Kahler metrics {gj}iEi glue up to a globally defined torsion-free linear connection D on M2n given by

DXY = VXY -

2

(w(X)Y + w(Y)X - g(X, Y)B)

(1.5)

1. L.C.K. MANIFOLDS

3

for any X, Y E T (M2n) . Moreover, D satisfies

Dg=w®g.

(1.6)

Proof. Let N be a C' manifold and g = exp(u)g two conformally related Riemannian metrics on N. Then the Levi-Civita connections t, v of §,g are related by

txY = OxY + 2 (X(u)Y + Y(u)X - g(X, Y)(du)')

(1.7)

for any X, Y E T(N). Let (M2n, J, g) be a l.c.K. manifold. Let us apply the previous considerations to N = U; and u = -f,. Then by (1.7)

DXY = VxY - 2 (w(X)Y +w(Y)X - g(X,Y))) for any X, Y E T (Ui ), so that the local connections Di glue up to a globally defined linear connection D (given by (1.5)). The verification of (1.6) is left to the reader.

The connection D furnished by Theorem 1.2 is the Weyl connection of the l.c.K. manifold (M2n, j, g).

Theorem 1.3 The Hermitian manifold (M2i, j, g) is a 1. c.K. manifold if and only if there exists a closed 1-form w on Mgr so that the complex structure J of M2n is parallel with respect to the connection D given by (1.5).

Proof. Note that g((DxJ)Y, Z) = (dSl)(X, JY, JZ) - (d 1)(X, Y, Z) for any X, Y, Z E T(M2n). If (M2n, j, g) is 1.c.K. then (1.3) yields

g((DxJ)Y, Z) = w(X)f (JY, JZ) - w(X)IZ(Y, Z) = 0. Conversely, the identities DJ = 0 and Dg = w ®g lead to D11 = w ®11, and then (1.3) follows from

(dIl)(X,Y, Z) = 1 > (DxI)(Y, Z), 6

xyz

where EXYZ denotes the cyclic sum over X, Y, Z.

CHAPTER 1. L.C.K. MANIFOLDS

4

Corollary 1.1 The Hermitian manifold (M2", J, g) is l.c.K. if and only if

OXJY = JOXY +2 {B(Y)X - w(Y)JX - g(X, Y)A - Q(X, Y)B}

(1.8)

for any X,Y E T(M2i). Here 0 = w o J and A = - JB are respectively the anti-Lee form and the antiLee vector field. In particular, (1.8) shows that l.c.K. manifolds belong to the class W4 (in the Gray-Hervella classification of almost Hermitian manifolds, cf. [1231).

Corollary 1.2 On each l.c.K. manifold VBJ = VAJ = 0. Next, we wish to demonstrate the connection between l.c.K. structures and conformal structures. To this end, let N be a C°° manifold. A conformal structure on N is a class G = {exp(A)g : A E COD(N)} of conformally related

Riemannian metrics on N where g is some Riemannian metric on N. A pair (N, G) consisting of a C°° manifold and a conformal structure is a conformal manifold. A Weyl structure on the conformal manifold (N, G) is T* (N) satisfying the property

a map F : G

F(exp(A)g) = F(g) - dA for any g = exp(A)g E G. The synthetic object (N, G, F) is a Weyl manifold.

Given a Weyl structure F on (N, G) the images F(g) of all metrics g E G have a common exterior derivative denoted by 4) and called the distance curvature of (N, G, F). On every Weyl manifold there is a unique torsionfree linear connection D the Weyl connection of (N, G, F), so that

Dg = F(g) ®g

(1.9)

for any g E G. Cf. G.B. Folland, [98]. Let (N, G, F) be an even-dimensional Weyl manifold. Assume that N admits an almost complex structure J so that g is an almost Hermitian met-

ric (hence any g E G is almost Hermitian) and DJ = 0. The fact that the Weyl connection D is torsion-less is then easily seen to yield the integrabil-

ity of J. Then (G, F, J) is a Hermite- Weyl structure and (N, G, F, J) is a Hermite- Weyl manifold.

1. L.C.K. MANIFOLDS

5

Theorem 1.4 Any Hermite-Weyl manifold of complex dimension n > 3 is 1. c. K. Conversely, any 1. c. K. manifold possesses a natural Hermite-Weyl structure with a vanishing distance curvature.

Proof. Let (M2n, G, F, J) be a Hermite-Weyl manifold. Let w = F(g). Let 11 be the 2-form associated with (J, g). Then

(dI)(X, Y, Z) =

6

(Dx0)(Y, Z) XYZ

1 > w(X)st(Y, Z) = (w n 11)(X, Y, Z) XYZ

so that (1.3) holds. Next, as Q is nondegenerate the mapping a'-- a A St"-2 furnishes a bundle isomorphism A2T*(M2n) -- A2n-2T* (M2n), provided that

n > 3. In particular, exterior multiplication with 0 is injective and then

0=d2S1=d(wAS1)=dwAQ-wA(wAQ)=dwAS1, hence dw = 0. Then, by Theorem 1.1, (g, J) is a l.c.K. structure and w its Lee form. Conversely, let (M2n, J, g) be a 1.c.K. manifold and w its Lee form. Define G and F : G roo(T*(M2n)) by setting G = {exp(A)g : A E COO(M2"))

and F(exp(A)g) = w - d A. Then F is a Weyl structure with 4> = 0 as w is closed, etc. Finally, let us mention the following (cf. I. Vaisman, [275]):

Proposition 1.1 Let (M, J) be a complex manifold and k its universal covering space. Then M admits a 1.c.K. structure if and only if k admits a Kdhlerian structure with respect to which G = 7rl (M) acts by conformal transformations. Proof. Let g be a l.c.K. metric on M whose associated Lee form is w. Then g lifts to Hermitian metric g on k with the Lee form Ca (the lift of w). As w is closed and k simply connected, ) is exact. Hence g is globally conformal

Kahler. Let then h = cpg (cp E C°°(M), cp > 0) be a Kahlerian metric. Now G acts by holomorphic isometries on (M, J, g). In particular, for any

aEG,a*g=g. Thus a*h

= a*(V9) =

o a)a*9 = (gyp o

so G acts by conformal transformations with respect to h. Conversely, let h be a Kahlerian metric on (M, J). By hypothesis G preserves the conformal class of h. Hence, by projection, one obtains a l.c.K. structure on M.

Chapter 2

Principally Important Properties 2.1

Vaisman's conjectures

A fundamental problem in l.c.K. geometry is to decide which l.c.K. manifolds admit some globally defined Kahler metric. We may state

Theorem 2.1 (Cf. [273]) Let (M2n,J,g) be a compact l.c.K. manifold. Then (M2n, j, g) is g. c. K. if and only if there is some global Kdhler metric on M2n.

Proof. Extend the Lee form w, by C-linearity, to T(M2) ® C and let w' (respectively w") be its (1, 0)-component (respectively its (0,1)-component). As w is closed we have

aw, =aw'=0 aw +aw"=0. Let 0 be the anti-Lee form. Then

0 = i(w' - w") and its exterior derivative is given by dO = 2ic9w

where i = V1_-1. Thus dO is a real exact (1, 1)-form. By the as lemma, if M2n

admits a Miler metric g then there is a CO° function f : M2n -' (0, +oo) so that dB = 2iaa log f.

(2.2)

8

CHAPTER 2. PRINCIPALLY IMPORTANT PROPERTIES

Define the metric h by setting h = fg. Then h is 1.c.K. Its associated fundamental 2-form is 11h = f 11. Then

dTlh = dfncl+fdcz = dfASl+fwA1l=(dlogf+w)AS2h. Thus the Lee form of (M2", J, h) is wh = w + d log f . From (2.2) it follows

that awh = aw, +'5a log f = as log f + as log f = o. Consequently dOh = 0. Yet, by (1.4), Oh

=-n118Slh

so that Oh is closed and coexact with respect to h, hence also coclosed with

respect to h. In particular it is harmonic. By the Hodge decomposition theorem it follows that 0h = 0 and consequently wh = 0, i.e. h is a Kahler metric. The converse is left as an exercise. The following sharper result actually holds (cf. P. Gauduchon, 1103]): A compact 1.c.K. manifold M is g.c.K. provided that its first Betti number bl is twice the irregularity q = dimc Hl (M, OM). In connection with Theorem 2.1 we may state the following conjectures of I. Vaisman:

Conjecture 2.1 Any compact 1. c. K. manifold satisfying the topological restrictions of a Kdhler manifold admits some global Kahler metric. The odd Betti numbers of a compact manifold which admits a Kahler metric are all even. Hence, the above conjecture suggests the following stronger one also motivated by the known compact examples:

Conjecture 2.2 Any compact l.c.K. but not g.c.K. manifold has an odd Betti number. Conjecture 2.2 was proved for 1.c.K. manifolds with parallel Lee form (cf. Chapter 5), i.e. if Vw = 0 then bl (M2i) is odd (cf. T. Kashiwada & S. Sato, [153], I. Vaisman, [275]). It is also true for compact complex surfaces.

Proposition 2.1 (Cf. [273]) A compact complex surface which admits a 1. c.K. metric but admits no Kahler metric has an odd first Betti number.

2.1. VAISMAN'S CONJECTURES

9

Proof. The proof relies on the following fact proved by K. Kodaira, [163]: If bl(M4) = 2q then there exist q closed, holomorphic, independent 1-forms ¢l, , (q so that their cohomology classes and the cohomology classes of their complex conjugates generate Hl (M4, C). It is crucial that M4 has complex dimension 2. Then we may write the Lee form as Its (1, 0) component is

w'= alol + -

-

- + aq(pq + aa.

Next, by (2.1), dO = 2iDw = 2it9aa. Proceeding along the lines of the proof of Theorem 2.1 one shows that M4 is g.c.K., a contradiction. The following results establish sufficient conditions in terms of curvature for a l.c.K. manifold to admit a global Kahler metric (cf. I. Vaisman, [269]).

Theorem 2.2 Let (M2n, J, g) be a 1. c. K. manifold. If its local Kdhler metrics gi have a definite Ricci tensor (e.g. if each gi is Einsten of nonzero scalar curvature) then M2n admits a Kahler metric. If additionally M2n is compact then it is a nonsingular projective algebraic variety. Proof. Clearly the Ricci tensors of the (local) Kahler metrics gi of M2n glue up to a (globally defined) tensor field SD on M2" (the Ricci tensor of the Weyl connection D of (M2", J, g)). As each gi is Kahler SD(X, Y) _ SD(X, JY) i' a 2-form. We have SD = as log(det gi); yet

detgi = det(exp(-f2)gIu;) = exp(-2nfi) detglu,,

so that SD = 2aalog(detg) - naw.

If s is the Ricci form of the metric g the previous formula may be written

SD = -v -1S - naw. As w is closed we have d(aw) = 0. Thus the first Chern class cl(M2i) is represented by -v/sD/(27r). Then SD/(27r) (or -SD/(27r)) is a Kahler metric on M. As to the last statement in Theorem 2.2, as cl (M2) is the image of an integer cohomology class and M2n is compact, then M2n is a Hodge manifold.

10

2.2

CHAPTER 2. PRINCIPALLY IMPORTANT PROPERTIES

Reducible manifolds

We proceed by recalling several facts on reducible R.iemannian manifolds. A m-dimensional Riemannian manifold M is locally analytical reducible if, with respect to some local coordinate system (Za, z6) on M, the metric may be written as ds2 = g,,p(z`)dzadzb + guz,(z')dz"dz°,

where the following conventions are adopted as to the range of indices:

a,b,cE

u,v,wE

and 1 < h < m - 1. Equivalently, there exist two analytical, complementary orthogonal foliations D and Dl so that M is locally the Riemannian product of a leaf of D and a leaf of Dl. Let P, Q be such that

P2=P, Q2=Q, PQ=QP=O

P+Q=I, g(PX,QY)=0 and

D. = Im(PP) , Dz = Im(Q-) for any x E M. If M is a complex manifold we additionally require that P, Q be holomorphic:

Pi = JP, QJ = JQ. In terms of P, Q the integrability of the distributions V and Dl is described by

Q[PX, PY] = 0, P[QX, QY] = 0. The following result is known (cf. I. Vaisman, 1267])

Proposition 2.2 A complex distribution V on a Kdhler manifold M is analytic if and only if [JX,Y] - J[X, Y] E V for any X E X(M) and Y E D. Proposition 2.2 has a local character, so it remains true on a U.K. manifold, as well. In terms of P, Q the two orthogonal complementary distributions V and Dl on the given l.c.K. manifold M are analytic provided that

Q{[JX, PY) - J[X, PY]} = 0

P{[JX, PY] - J[X,QY]} = 0. Next g is locally a product metric if

(Lpxg)(QY,QZ) = 0

2.3. CURVATURE PROPERTIES

11

(LQxg) (PY, PZ) = 0. Assume now that the l.c.K. manifold (M, J, g) is analytically reducible. The previous identities yield (dfz)(PX, PY,QZ) = 0 (dc2) (QX, QY, PZ) = 0

or (bydfl=wAIl) w(PZ)SZ(QX, QY) = 0

w(QZ)St(PX, PY) = 0

from which, by conveniently choosing X and Y, w = 0. We proved the following (cf. I. Vaisman, [271]).

Theorem 2.3 Any analytically reducible l.c.K. manifold is Kahler.

2.3

Curvature properties

Let us now recall A. Gray's classification (cf. [118]) of Hermitian manifolds: M E (i) if the curvature tensor R of M satisfies the identity (i), where (1)

R(X,Y,Z,W) = R(X,Y,JZ,JW)

(2)

R(X, Y, Z, W) = R(JX, JY, Z, W) +R(JX, Y, JZ, W) + R(JX, Y, Z, JW) R(X, Y, Z, W) = R(JX, JY, JZ, JW).

(3)

The intersection of these classes with that of l.c.K. manifolds is rather small. Precisely, we have

Theorem 2.4 (Cf. [271]) Let M be a l.c.K. manifold. Then a) if M is compact or div(B) < 0 and M E (1) then M is Kahlerian, b) M E (2) if and only if M E (3). If this is the case, and additionally B is analytic or the trajectories of A and B are geodesics, then M is Kahlerian. Proof. Let us consider the following (0, 2) tensor fields:

L=Vw+2w®w

L=L-Lo(J,J).

CHAPTER 2. PRINCIPALLY IMPORTANT PROPERTIES

12

The curvature tensor fields of g and g' = exp(-a)g are related by

exp(a)RD(X,Y,Z,W) = R(X,Y,Z,W) 2 {L(X, Z)g(Y, W) - L(Y, Z)g(X, W)

+ L(Y, W)g(X, Z) - L(X, W)g(Y, Z)} II 4112

(2.3)

{g(Y, Z)g(X, W) - g(X, Z)g(Y, W)}

Assume M E (1). Using (2.3) and the fact that

RD(X,Y, Z, W) = RD(X,Y,JZ,JW)

as RD is the curvature tensor of a (local) Kfhler metric, after a suitable contraction, we find 211L112 = nIIwI12

Also, from the definition of L: 1IL112 = div(B) + 111w112

Thus 2div (B) = (n -1)IIw112.

Thus w = 0 (i.e. M is Kahlerian) if div(B) < 0 or M is compact. In this case one integrates over M and uses Green's lemma. This proves a). As to b), we find the following identity characterizing the class (3):

g(X,Z)L(Y,W) - g(Y,Z)L(X,W)

- g(X, W)L(Y, Z) + g(Y, W)L(X, Z) = 0. A suitable contraction leads to L = 0, characterizing (3). Similar considerM E (3). The converse is obvious. Now, for ations show that M E (2) M E (3) one finds g((,CBJ)X, JY) + 2 {w(X)w(Y) - 8(X)e(Y)} = 0.

If B is analytic, then w ®w = 0 ® 0 and thus w = 0. Finally, if X = B we get

IA, BI = 2 (Iw1I2A etc.

2.3. CURVATURE PROPERTIES

13

Remark 2.1 While is has been demonstrated that a 1.c.K. metric g is not necessarily Kahler when its curvature tensor possesses the symmetries of the curvature tensor of a Kahler metric (i.e. when M E n 1(i)), one may show that a compact l.c.K. manifold is Kahler if and only if the curvature tensor of the Chern connection possesses the symmetries of the curvature tensor of a Kahler metric, cf. [271]. The next results deal with the holomorphic sectional curvature of a l.c.K. manifold. Using a result of A. Gray, [118], we get (cf. I. Vaisman, [271]).

Proposition 2.3 Let M be a complete l.c.K. manifold whose holomorphic sectional curvature k(X) = K(X, JX) satisfies

k(X)>

2

II4II

+6, 6>2.

Then M is compact and simply connected, hence g. c. K.

We may also state (cf. I. Vaisman, [271]): Theorem 2.5 Let M be a connected l.c.K. manifold all of whose local Kahler metrics have the same constant holomorphic sectional curvature k0. Then ko = 0 or M is g.c.K. In the first case (ko = 0) if additionally g has pointwise constant holomorphic sectional curvature different from -IIWII2/4 at one point of M at least, or M is compact, then M is g.c.K. Proof.

Let us replace RD in (2.3) by the well known expression of the

curvature tensor field of a complex space form. By suitable contraction we obtain

n(n + 1)ko exp(-o) = s + 2(n -1)IIL112 -

n(2n 2-1) II,II2.

Suppose ko = 0. Then

k(X) = 1 IIwII2 + 1(L(X, JX) - L(JX,X)) 4

2

for any unit vector X. Thus k(X) does not depend upon X if and only if

L(X, JX) - L(JX, X) = cpg(X, X)

for arbitrary X and some cp E C°O(M). Using the symmetry of L, by polarization we get

L(X, JY) - L(JY, X) =

Y).

CHAPTER 2. PRINCIPALLY IMPORTANT PROPERTIES

14

On the other hand

L(X, JY) - L(JY, X) _ (Vxw)JY - (Vjyw)X +2(w A0)(X,Y) = (de)(X,Y) + 2(w A0)(X,Y)

(LBI-(d0))(X,Y) = (IIII2

-

AG) + (do)) (X,Y)

and thus

dO - 2 (w A 0) = ?PSt

where V) =-IIw112-W.

Let us differentiate (2.4) to find (as rank(Q) = 2n) 2d?p +*w=0.

As w is locally exact, set w = 2d log r. Then .0 = c/T (locally) for some c E R. As M is connected, either -0 is identically zero, or it is nowhere vanishing. In the last situation w is exact and M is g.c.K. The first case actually does not occur because otherwise k(X) = -11W112 /4, a contradiction. Finally, if M is compact and 7P = 0 then (2.4) becomes 2d0 = w A 0 hence 2d611 = w A 0. Therefore 2 (652, 612) = (W A 652,12) = - n 1 1(652, 6S2).

Consequently 612 = 0 hence 0 = 0. The proof is complete.

Proposition 2.4 (Cf. [271]) If M is a compact l.c.K. manifold whose curvature tensor satisfies R(X, JX, Y, JY) > R(X, Y, X, Y) + R(X, JY, X, JY)

then M is Kdhlerian. Proof. Let X = E, and Y = JE, and sum over i in the above identity, where {Et} is a (local) orthonormal frame. Then

Hq -R->0

(2.5)

where H(X,Y) = H(X, JY) and H is the (0,2) tensor field of local components H, = (1/2)Rjjr,Jr,. As it has been shown by T. Kashiwada, [150], the inequality (2.5) actually implies that M is Kahlerian.

2.4. BLOW-UP

15

Note that (2.5) holds good if M has constant sectional curvature. On the other hand, as there are no complex structures on higher dimensional spheres, M cannot have positive sectional curvature. To be sure, one has actually to exclude the case dimc M = 3. Proposition 2.5 A compact l.c.K. manifold of constant sectional curvature is a flat Kahlerian manifold. Cf. I. Vaisman, [271].

Remark 2.2 A compact I.c.K. (but not g.c.K.) manifold cannot have strictly positive sectional curvature because (by a classical result of J. Synge, [248]) it would be simply connected and thus g.c.K.

2.4

Blow-up

We now present two results by F. Tricerri (cf. [258]) which show that the l.c.K. category is closed under 'blowing-up,' the second of which (cf. Theorem 2.7 below) implies that, as long as one deals with complex surfaces alone, it suffices to look at minimal 1.c.K., not g.c.K., compact surfaces lying in the classes VIo and VII0 of Kodaira's classification.

Theorem 2.6 The blow-up at a point of a I.c.K. manifold admits a l.c.K. structure. Proof. Let (M, J, g) be a I.c.K. manifold and x E M. We firstly find an open neighborhood U of x and a global metric g", conformal to g, and thus I.c.K. too, and such that the restriction of g" to U is Kahler. This can be done as follows: Consider on open neighborhood V on which g = exp(h)g' for some local Ki hler metric g'. Then take U a relatively compact open neighborhood

of x so that U c V' C V (V' open). Now let f E COO (M) be 1 on U and

vanishing on M - V. Let h E C' (M) so that h = for on V and h = 0 outside. Then the required metric is g" = exp(h)g. We look at the Killer manifold (U, gnu, J). As it is Kiihler, its blow-up U at x admits a Killer metric g (cf. M. Berger & A. Lascoux, [21]). The blow-up M is obtained by glueing together M - {x} and U. As g agrees with g" on the common part, we obtain a I.c.K. metric on k which is Killer on U. The converse is also true.

Theorem 2.7 If the blow-up at a point of a complex manifold admits a I.c.K. metric, then the manifold itself admits a Lc.K. metric.

16

CHAPTER 2. PRINCIPALLY IMPORTANT PROPERTIES

Proof. Let M be a complex manifold, p E M, and k the blow up of M in p. Let 7r : M -+ M be the natural projection. Notice first that restricting 7r to M - 7r-1(x) one obtaines a biholomorphism onto M - {x}. Thus a 1.c.K. metric g on M induces a l.c.K. metric g on M - {x}. The problem is to extend it at x. To this end we establish the existence of an open neighborhood U of x so that 91M-{x} = exp(f)g'

with f E C' (U - {x}) and g' a Kahler metric on U - {x}. Take U to be an open coordinate neighborhood of x homeomorphic to D = {z E C" : IzI < r}. Then one may build a deformation retract of 7r-1(U) onto a-1(x) CPr-1. Thus 7r-1(U) is simply connected and the restriction of the Lee form w to it is exact; in turn, this shows that the restriction of g to 1r-1 (U) is g.c.K. by means of some f E C°°(7r-1(U)). Then the required f is f'o7r-1. Once the pair (U, f) is constructed we can assume (cf. the previous proof, eventually shrinking U) that 91U-{x} is Kahler. Finally one may apply a result of Y. Miyaoka, [190], to ensure existence of an open neighborhood V c U and of a Kahler metric on U which agrees with g on U - V. This completes the proof.

2.5

An adapted cohomology

Let us describe an adapted cohomology available on a l.c.K. manifold (cf. I. Vaisman, 1272]). Let us consider the operator

d,,=d+e(w) acting on forms where e(w) denotes exterior multiplication by w. Clearly squares to zero. Let HP(M) be the p-th cohomology group associated with the resulting cochain complex. Let F,, (M) be the sheaf of germs of smooth

real functions f on M with d 0 is a germ in .P,,,(M) if and only if f g is the germ of a Kahler metric on M. Now we may prove the following (cf. I. Vaisman, [272]).

Theorem 2.8 The groups HA(M) are isomorphic to the cohomology groups of M with coefficients in the sheaf F,, (M). Proof. Let a be a local d,-closed form. By eventually shrinking the open set U on which a is defined, we may assume that w = b-1db on U, b 96 0. Then 4(b-1Q) for some local form /3. Thus 4 satisfies the d(ba) = 0 so that a =

2.5. AN ADAPTED COHOMOLOGY

17

Poincare lemma. Consequently, letting AP(M) denote the sheaf of germs of differentiable p-forms on M, one may check that 0

.F,(M)

A°(M)

A'(M)

...

is a fine resolution of .7 (M) and the proof is complete.

We end this section by presenting a result by F. Guedira & A. Lichnerowicz, [117]. Let (M2n, J, g) be a l.c.K. manifold whose Lee form w is not exact. We denote by g* the pointwise inner product and by (, ) the L2 inner

product induced by g on p-forms on M. If B is the Lee field, we introduce the differential operator of degree -1 on p-forms defined by bw = S + LB.

where 6 = d* is the adjoint of d with respect to and tX denotes the interior product with the vector field X. Let a be a (p - 1)-form on M and a p-form. It is known that 9*(e(w)a>)3) =9*(a,LB)3)-

It follows that (a, SWQ),

provided that the intersection of the carriers of a and /3 is compact. The operator 5,, satisfies (6,)2 = 0. We shall need the following:

Lemma 2.1 On any connected l.c.K. manifold M whose Lee form w is not exact any top degree 5,-closed form vanishes. Proof. Indeed, any 2n-form on M may be written f * 1, where f E C°°(M). Let V be the Levi-Civita connection of (M, g). With respect to a local coordinate system (U, xi) on M the components of b,,,(f * 1) are

-(V f * 1 is 5,,-closed if and only if

df-fw=0. Let us study the zero set of f on M. By (2.6) this set cannot be empty, because otherwise w would be exact, a contradiction. Then let {Ui} be a locally finite open cover of M so that each Ui is contractible. One has

CHAPTER 2. PRINCIPALLY IMPORTANT PROPERTIES

18

Wlu, = V ldtpi for some C°° function Vi : U; -, (0, +oo). On Ui the identity (2.6) becomes dflu; - fiu.ci 1dco2 = 0,

that is d(W 'f,U.) = 0. Hence we have

flu, = kici on U,, for some ki E R. Therefore, if f is zero at a point of Ui then it vanishes on the whole of Ui. It follows that the (nonempty) zero set of f is both closed and open. Yet M is connected, so this zero set coincides with M, hence the given 6,,-closed 2n-form vanishes identically, Q.E.D.

Let us consider on M the generalized Laplacian AW = db,,, + bwdd

whose principal part is the ordinary Laplacian A of (M, g) and which is selfadjoint with respect to g. If the intersection of the carriers of the p-forms a and /3 is compact, then we have: (A ,a, /3) = (a, A.#) = (dwa, dj) + (6wa, 6.0)

Let us assume that M is compact. By following the ideas of G. De Rham, [71J, p. 153-154, one may easily check that each &-closed p-form on M is dWhomologous to a p-form which is both &-closed and b,,-closed. In particular,

any top degree form on M is &-homologous to a 6,,,-closed 2n-form, hence vanishing, by Lemma 2.1. Thus any top degree form on M is &-exact. If M is noncompact, let /3 be a L2 top degree form on M, that is (Q, a) < Co.

Again by following the ideas in [711, p. 169, one may show that,3 is 4-exact.

Let now a be an arbitrary top degree form on M. Then: (exp(- f )a, exp(- f )a) < oo

for some function f on M, that is /3 = exp(- f )a is a L2 form. Consider the following closed (yet nonexact) form

77 =w+df.

2.5. AN ADAPTED COHOMOLOGY

19

Set g f = exp(f)g. Then g1 is a 1.c.K. metric on (M, J) and 17 is its Lee form. Let 4 = d + e(rl) be the corresponding cohomology operator. As a is a top

degree L2 form, it is d,,-exact and there is a (2n - 1)-form o on M so that 0 =d,70=dV)+77 AV).

Therefore,

a=exp(f)di,b+r,Aexp(f)Vi=d(exp(f)-i)+(77 - df ) A exp(f)*, Hence by the very definition of r)

a = &(exp(f )iii) and the 2n-form a is dW-exact. We have proved the following (cf. F. Guedira & A. Lichnerowicz, [117]):

Theorem 2.9 Let M2n be a connected l.c.K. manifold whose Lee form w is not exact. Then H,,n(M) = 0.

Chapter 3

Examples 3.1

Hopf manifolds

Let A E C, 1a[ # 1, and let A. be the cyclic group generated by the trans-

formations z i-* Az of C" - {0}. Then Aa acts freely on C" - {0} as a properly discontinuous group of complex analytic transformations (cf. S. Kobayashi & K. Nomizu, (1621, vol. II, p. 137). Thus the quotient space CH" = (C" - {0})/Da has the structure of a complex manifold. This is the complex Hopf manifold. Let

f:

C"-{0}-+RxS2n-1

be the diffeomorphism given by z H C7r exp(

2

[zz1)

log

.

If A = e2 then f commutes with the actions of Aa and Z on C" - {0} and R x S2n-1, respectively. Thus f induces a diffeomorphism F : CHa S1(1/a) X S2n-1 so that the following diagram is commutative:

C"-{0} 7r l

CHa = (C" - lo})/A,\

RxS2"-1

!p (RxS2n-1)/Z

S1(1/lr) X

S2n-1

Here it and p are the natural covering maps. In particular CHa is compact

and its first Betti number is b1(CHa) = 1. Thus CHa admits no global Kahler metrics, provided n > 1. Hence the result in [81 is wrong.

CHAPTER 3. EXAMPLES

22

The Hermitian metrics (on C" - {0}) 9t = 4

zI2tzJ 6jkd

®dz-k

+ tIzI2(t-1)(E zj dzl) ®(E zkdz) Iz12(t+1)

I

(3.1 )

for t E [-1, oo), are DA-invariant, thus inducing global Hermitian metrics on CH.'. The Kahler 2-form of gt satisfies d1 = wt A lit with wt=-(1+t)E(ddz-j z?dzi)

which is not exact for t # -1. Two cases are of particular interest: i) If t = 0, the corresponding metric 6jkdz3 ® dxk go

IZI2

is globally conformal Kiihler on Cn - {0} with the Lee form

(zJdzz+z?dz?)

wo =

(3.3)

The induced metric go on CHa is called the Boothby metric (it was discovered

by W. M. Boothby, [36], for n = 2). Endowing R x Sii-1 with the standard product metric which is Z-invariant, one may check that f is an isometry. Thus F is an isometry of (CHa, go) and S' (1/ir) X S2n_1 with the product metric. Consequently CHn\ is CO0 reducible. If we let dt2, respectively dv2, be the length element on S1, respectively the standard metric on 52n-1, then go and wo satisfy go = f *

(-t2 + do2J

wo=

-2f*dt.

One may check that wo is parallel and has constant length IIwoII = 2. As we shall see later on, this observation leads to an important generalization of Hopf manifolds, namely to the class of generalized Hopf manifolds. ii) If t # 0 the Lee vector field is expressed by BL

2

\.

8zi

+ zj tU

/

which remarkably does not depend on t. It may be shown that it fails to be parallel with respect to the Levi-Civita connection of gt.

3.2. THE INOUE SURFACES

23

If N is a compact projective variety, then the total space of the induced Hopf fibration carries a 1.c.K. metric with parallel Lee form (cf. I. Vaisman, (273]).

We should mention that no attempt to construct l.c.K. metrics on other Hopf surfaces (e.g. H,,,p with Ial 76 I/6I, cf. 113]) has proved successful, as yet. The existence of l.c.K. metrics on Hopf surfaces other than CHa is an open problem. The metrics gt appear for the first time in [275].

3.2

The Inoue surfaces

These are quotient surfaces having the covering space H x C where H denotes

the upper halfplane {w = w1 + iw2 E C : w2 > 01. To construct them we start (cf. M. Inoue, [141]) with a unimodular matrix M = [msj] E SL(3, Z) having one real eigenvalue a > 1 (this actually implies that a is irrational) and two complex conjugate eigenvalues Q : Q. Let (al, a2, a3)t be a real eigenvector corresponding to a and (b1,b2ib3)t an eigenvector for 0. Let GM be the group of complex analytic transformations of H x C generated by fo : (w, z) '-+ (aw, Qz)

(3.4)

fj : (w, z) F--* (w + aj, z + bj) , j E {1, 2, 3}.

(3.5)

It may be shown that GM acts freely on H x C as a properly discontinuous group of complex analytic transformations so that the quotient space SM = (H x C)/GM becomes a (compact) complex surface. We wish to compute its Betti numbers. To this end we compute the abelianized group GM/[GM, GM]. The generators (3.4)-(3.5) satisfy the relations

fjfk = fkfj , k, j E {1,2,3} fOfjfO 1 =

f1"-11

f2 ,2 f3 j3

Then GMl [GM, GM] is generated by 70,-..,73 with relations 3

E(mkj -bkj)fj =0, kE (1, 2,31. j=1

Thus 3

GMl[GM,GM] - Z4/(>(mkj - bkj)fj) j=1

CHAPTER 3. EXAMPLES

24 3

Z ® Z3/(E(mkj - bkj)ej), /

j=1

1(mkj - bkj)ej are inwhere ej are the generators of Z3. The elements dependent over Z by the properties of M. Thus, by the theorem of invariant factors GM/[GM, GMT ti Z ®Zdl ®Zd2 ®Zd3,

where dj are the invariant factors of the matrix M - 13. Then the free part of GM/[GM,GM] is Z hence dimR,H1(SM) = 1 and bl(SM) = 1. Thus Sm admits no global Kahler metrics. On the other hand, we may consider the Hermitian metric dw®dw (3.6) w22 +w2dz ®dz g=

on H x C. This will be referred to as the Tricerri metric. For its Kahler form we find d Q = d log w2 A S2

with an exact Lee form w = d log w2. Thus (3.6) is g.c.K. Note that (3.6) is GM-invariant, so that it induces a global l.c.K. metric on SM. We shall now follow F. Tricerri [2581 to compute the covariant derivative of w. As our considerations are local in character, we may as well perform all calculations on H x C. It is convenient to use real coordinates and consider the orthonormal frame E1

e

a

8

1

= w2 8a , E2 = W2 8w2 , E3 =

C

' E4 =

1

8

w2 8z2

Note that E2 = B = wp. The dual coframe will be 01 = dw1 , 02 = dw2 W2

W2

03 =

-2dzl , 04 =

w2dz2.

Letting 0ti be the connection 1-forms, Cartan's structure equations yield

dO1=01A02,d92=0 2d93=02A03,2d04=02A04. Thus 1

31 3

2 03=-92--28 = -2 = - B2B1_01,

= 194 4=-042--2

02

3.3. A GENERALIZATION OF THURSTON'S MANIFOLD

25

and the remaining components of the connection matrix are zero. On the other hand

V E2 = VE;B = ez(E.i)Ek

= b El - 2 b E3 - 2 S E4. Thus DE; B # 0 for j 34 0 so that the Lee form is not parallel, yet it may be easily seen to be harmonic.

Note that [JB, B] _ -B so that the complex distribution generated by B and JB is integrable. However, it is not regular because the Inoue surfaces have no complex curves. Finally, we mention that other Inoue surfaces possess natural l.c.K. metrics, as well (cf. F. Tricerri, [258]).

3.3 A generalization of Thurston's manifold Let H(r, 1), r > 1, be the generalized Heisenberg group consisting of all matrices with real entries of the form

a= where A = (al, .

,

a,), B = (bi,

1

AC

0

Jr Bt

0

0 ,

1

b,.) E R. Then H(r, l) is a simply

connected, compact nilpotent Lie group of real dimension 2r + 1 (cf. [127]).

Let r(r, 1) C H(r, 1) be the subgroup of all matrices with integer entries. Set

N(r,1) = H(r, l)/I'(r,1). This is a compact manifold. Next we endow H(r, 1) the global coordinate chart (x', yi, z) given by

x'(a) = ai , y (a) = bi, z(a) = c, i E { 1,

,

r}.

A frame of left invariant 1-forms will be

a'=dx',/3 =dy',ry=dz-Ex'dyi with the corresponding dual frame of left invariant vector fields a

a

a

a

Xi=axi'1`=ayi+az'Z=Oz

CHAPTER 3. EXAMPLES

26

We endow H(r, 1) the left invariant metric

g = E(a`0ai+/3`®Q`)+7®7 with respect to which the previous frames are orthonormal. The 1-forms, vector fields, and metric constructed above project on N(r, 1) via the natural N(r, 1). For simplicity we shall denote their images by projection H(r, 1) the same symbols. Consider the product manifold

NS(r,1) = N(r, l) x Sl. As bl(NS(r,1)) = 2r + 1, these manifolds admit no global Kiihler metrics. On the other hand, the product metric g = g+dt2 is 1.c.K. (cf. L. A. Cordero & M. Fernandez & M. De Leon, [68]) with respect to the natural complex structure. Set T = Then {Xi,Y,Z,T} is an orthonormal frame with respect tog and (ai, f3i, -y, dt} is the corresponding coframe. We define the almost complex structure J by

JXi=Y,JY=-Xi,JZ=T,JT=-Z. A computation shows that all brackets of the fields in the frame vanish except

[Xi,Yj] = Z when i # j. Thus J is a complex structure. With respect to the chosen coframe the Kahler form reads

Q=E(a&A/3i)+7Adt and one may check that dS2 = dt A Sl. The Lee form w = dt is easily seen to be parallel.

3.4 A four-dimensional solvmanifold We follow L. C. De Andres & L. A. Cordero & M. Fernandez & J. J. Mencia, [6]. This example provides a family of compact l.c.K. manifolds with nonparallel Lee form as the total space of a circle bundle over a three-dimensional

solvmanifold M3(k). We first describe the construction of M3(k), cf. L. Auslander & L. Green & F. Hahn, [9]. Let G(k) be the group consisting of all matrices of the form 1ekz

A=

0 0

0

0

0x

e-kz 0 g 0 0

1

z

0

1

3.4. A FOUR-DIMENSIONAL SOLVMANIFOLD

27

where x, y, z, k E R and k is such that ek + e-k E Z - {2). Then G(k) is a connected solvable Lie group admitting the global coordinate functions

x(A) = x, y(A) = y, z(A) = z. A basis of right invariant 1-forms is

dx - kxdz , dy + kydz , dz.

By a result in [9] there exist discrete subgroups of G(k) such that the corresponding quotient is compact. Let r(k) be such a subgroup and set M3(k) = G(k)/r(k). Let a,,3, -y be the projections on M3(k) of the 1-forms in the above basis of right invariant forms on G(k). Let go be the metric on M3(k) given by

go=a2+ /32+72 The real cohomology of M3(k) is also known (cf. [9]):

H°(M3(k),R) _ {1}

H'(M3(k),R) _ {[y]} H2 (M3 (k), R) = {[a A#]}

H3(M3(k), R) = {[a A /3 A'y]},

thus the Betti numbers equal 1. Choose A E Z so that Ala A /3] is an integer cohomology class. As the circle bundles over M3(k) are classified by H2(M3(k), Z), for each integer n there is a principal circle bundle

it : M4(k,n) , M3(k) associated to nA[a A,3]. Moreover there is a connection 1-form 77 whose curvature form is nAa A,3. An easy computation shows that the Betti numbers of the total space equal 1 except for b2 = 0. Thus M4(k, n) admits no global

Kahler metrics. We build a l.c.K. metric on it as follows. Let {X, Y, Z, T} be the frame dual to the coframe {a, /3,'y, 71) and define the metric

g=a2+)32+'y2+7/2. We define an almost complex structure J by

JX = (nA/k)T, JY = Z, JT = -(k/nA)X, JZ = -Y. Then J is integrable and g is Hermitian. The Kiihler form of (J, g) is n = (nA/k)a A 77 +,3 A'y

and the equation di l = w A S2 is satisfied with w = -k-y. Moreover Vw 3k 0 so that M4(k,n) is l.c.K. (for n 0 0) with a nonparallel Lee form. In [6] an explicit realization of M4(k,n) is also given.

CHAPTER 3. EXAMPLES

28

3.5

SU(2) x S1

The authors are grateful to P. Gauduchon for the following example. Let V, X, I be the natural basis of left invariant vector fields on S = SU(2). Then IV, X] = 21, [V, I] = -2X, [X, I] = 2V. One defines a Riemannian metric g, on S by setting

gµ(V,V) = 1, g"(X,X) =gµ(I,I) =µ2. With this structure, S is called a Berger sphere. Now let B be a nonzero tangent vector field on S' and let J be the almost complex structure on S X Sl given by

JB=V,JX=I,JV=-B,JI=-X. It is straightforward that J is integrable and

VBJ=0,VvJ=0 (VxJ)B = -(1/µ2)I , (VxJ)V = -(1/p2)X

(VxJ)X = V , (VxJ)I = -B. Thus the Lee field is -(2/µ2)B and it is parallel.

3.6

Noncompact examples

Noncompact examples of 1.c.K. manifolds are not abundant in the literature.

I. Vaisman constructed (cf. [266]) a l.c.K. metric on the product C" x T. J.C. Marrero & J. Rocha have recently constructed new examples in [178]. There, the product of a c-Sasakian manifold with a c-Kenmotsu manifold, a typical example of the latter is any warped product R x f F with F Kahler and f = exp(t), is endowed with the standard complex structure assigned to the product of two contact metric manifolds and the product metric. Then one may check that the resulting structure is l.c.K. and its Lee form is the pullback of the contact form of the given c-Kenmotsu manifold. As R2"+1 has a natural c-Sasakian structure, new noncompact examples may be produced. However, it seems to be difficult to find noncompact examples which admit no global Kahler metrics.

3.7. BRIESKORN & VAN DE YEN'S MANIFOLDS

3.7

29

Brieskorn & Van de Ven's manifolds

In their attempt to construct complex structures on products S' x L, where S1 is the unit circle and L an odd-dimensional homotopy sphere, E. Brieskorn & A. Van de Ven, [45[, have generalized complex Hopf manifolds as follows. Let n > 1 and (bo, , bn) E Zi+1 , bj > 1, 0 < j < n. Let (zo, . , zn) be

the natural complex coordinates on C"+1 Define X2n(b) = X2n(bo, ... , bn) c Cn+1 by the equation (zo)bO+...+(zn)b° =0.

Then X2n(b) is an affine algebraic variety with one singular point at the origin of Cn+1 if bb > 2, 0 < j < n, and without singularities if bj = 1 for at least one j. Next B2n(b) = X2n(b) - {0}

is a complex n-dimensional manifold, referred hereafter as the Brieskorn manifold determined by the integers bo, , bn. Cf. also [441. Let a E C, 0 < jal < 1, be a fixed complex number. There is a natural holomorphic action of C on B2n(b) given by twa

t(zo,...,za)= (zoexP (___) ,...,znexp

-twa

bn

(3.7)

where t E C and wa = - log jal - i arg(a). Then Z acts freely and properly discontinuously on B2n(b) as a subgroup of C. Consider the complex manifold

Ha (b) = B2n(b)/Z.

Note that H ( 1 , . . . ,1) and (C" - {0})/Da are diffeomorphic. Let D1 be the punctured open unit disk in C. Consider

f : D' x B2n(b) - Dl x B2n(b) defined by f (a, x) = (a, Uax), for any a E Dl , x E B2n(b), where Ua E

GL(n + 1, C) is the matrix(_

\

/

Ua=diag expbo 1,...,expl-bnl/ /x

Note that f is an automorphism of D1 B2n(b). The action of GL(n + 1, C) on Cn+1 induces an action of Z -_ { f'n : m E Z} on Dl x B2n(b). Let

Yn = (D' x B2n(b))/Z be the quotient space. The following result holds (cf. [771):

CHAPTER 3. EXAMPLES

30

Theorem 3.1 Y" is a complex n-dimensional manifold. Moreover, if n = 2 then there exists a surjective holomorphic map 7r : Y2 D' which makes y2 into a complex analytic family of compact complex surfaces. For any a E D1 there is a diffeomorphism between it-1(a) and HQ(b).

We recall that a triple (Y, ir, M) is a complex analytic family of compact complex manifolds if Y, M are complex manifolds and it : Y -+ M is a proper holomorphic map of maximal rank at all points of Y. Then each fibre 7r-1(a), a E M, is a compact complex manifold. Note that the action (3.7) of Z on B2n(b) generalizes slightly the one in [451, p. 390. There B2n(1, . , 1)/Z is diffeomorphic to (Cn - {0})/01/e. The proof of Theorem 3.1 is reminiscent of [1971 and is too technical to be reproduced here. The proof is organized in several steps, as follows. First, it is easy to show that Z acts freely on D' x B2n(b). Next, one shows that { fm : m E Z} is a properly discontinuous group of analytic transformations of D1 x B2n(b). Thus Yn is a complex manifold. Let p : D' x B2n(b) -+ Yn be the natural surjection. Moreover, let i : D1 x B2n(b) -+ D1 given by *(a, x) = a. As frr o f = * there exists a map it : Yn --+ D1 so that 7r op = fr. Asp is a covering map, it follows that 7r is surjective, holomorphic and of maximal rank at all points of Y. Finally, the last step is to show that it : Y2 -+ D' is a proper map. See [771 for details.

Let CHQ = (Cn - {0})/Aa be the complex Hopf manifold (cf. Section 3.1) and define Qn-1 C CHQ by setting Q.-1 = {ir(Z1, ... , z') : (z1)2 +

... +

(z'n)2

= 0}

where it : Cn - {0} -i CHQ is the natural covering map. Note that (z1)2 + + (z")2 = 0 is Aa invariant, so that Qn-1 is well defined. It is a complex

hypersurface of CHQ and it is referred to as the complex sphere in CHa. Let b3 E Z, b,, > 0, 1 5. Then the complex sphere Qn-1 in CHQ is a compact l.c.K. manifold which admits no globally defined Kahler metrics.

3.7. BRIESKORN & VAN DE VEN'S MANIFOLDS

Proof. Define f : CH." -+ S' X f : ir(z)

31

S2n-1 by setting

x1 . (3.8) loga ) ' xi l Note that the right-hand member of (3.8) is A, ,-invariant so that f (ir(z)) is well defined. Then f is a diffeomorphism with the obvious inverse Cexp

f-1 : (w,() for any w E S', C E

/27ri log I z 1

7r(aar8(w)/27r0

(3.9)

S2n-1. Then (3.8) induces a diffeomorphism:

Qn-1

S1 X L2"3(2,. .. , 2).

(3.10)

By Corollary 2.10 in [74], p. 58, it follows that Hi (Qn-1; Z)

Hj-1(L2n-3; Z)

®Hj (L2n-3; Z)

(3.11)

L2n-3(2, , 2). Set j = 2 in (3.11); by a result of where L2n-3 is short for E. Brieskorn, [441, one has Hi(L2n-3; Z) = 0 , i = 1, 2.

provided that n > 5. Thus H2(Qn_1; Z) = 0. Let G be any abelian group. L2n-3 is connected, by the universal coefficient theoSet j = 1 in (3.11); as rem, it follows that H2(Qn_1i G)

Tor(Z, G) = 0.

Assume further that G is a principal ideal domain (e.g. Z or a field). Then H2(Qn_1iG) Hom(H2(Qn_1iG),G) = 0 (cf. e.g. [189], p. 259). In particular H2(Qn_1i R) = 0. Therefore Qn_1 is a compact (by (3.10)) complex manifold which carries no globally defined Kahlerian metric. Clearly Qn_1 inherits a l.c.K. structure as a complex submanifold of CHa.

Remark 3.1 1) Note that Qn_1 = H. -1( 2 ,--

, 2). E. Brieskorn & A. Van de Ven (cf. [45)) show that each product between S1 and a (2n-1)-homotopy sphere, n 0 2, bounding a parallelizable manifold carries a complex structure

in a natural way. If bj = 2, 1 < j < n, then Theorem 3.2 completes this result, i.e. puts a Riemannian metric on H1(2,. . , 2) which is l.c.K. with respect to the complex structure discovered in [45]. 2) One may show that b2(L2n-3) = 0 using results from differential geometry as follows. By a result of Y. Tashiro, [2531, L2n-3 inherits an almost contact metric (a.ct.m.) structure, as a real hypersurface of the Kahlerian manifold B2n-2(2,---,2). Next, cf. S. Sasaki & C.J. Hsu, [242], this a.ct.m. structure is actually Sasakian. Finally, one may use Proposition 1 of S.I. Goldberg, b2(L2n-3) = 0. [108], p. 106, to conclude that

Chapter 4

Generalized Hopf manifolds The first example in the preceding section led to the study of locally conformal Kahler manifolds with a parallel Lee form. These are called Vaisman manifolds. I. Vaisman, to whom the notion is due (cf. [269] and [2751) adopts the terminology generalized Hopf (g.H.) manifolds. However, the term g.H. manifold is sometimes used to label a different generalization of complex Hopf manifolds (i.e. E. Brieskorn & A. Van de Ven's g.H. manifolds I Q`(b), [45], cf. our Section 3.7). In the following, we use both terminologies interchangeably. The example of an Inoue surface with the Tricerri metric shows that g.H. manifolds form a proper subset of the set of all l.c.K. manifolds.

The interest for the study of these manifolds is also motivated by the following:

Theorem 4.1 Let (M2,,J,g) be a compact l.c.K. manifold. If the local Kahler metrics of M have nonnegative Ricci curvature then g is globally conformal to a l.c.K. metric with parallel Lee form. Cf.

I. Vaisman, [275]. The key ingredient in the proof is a result by P.

Gauduchon (cf. [101]):

Lemma 4.1 On each compact Riemannian manifold, for any 1 form w there is a C°O function h so that w + dh is coclosed with respect to the metric exp(h)g. Proof of the theorem. Let h be as in Lemma 4.1 applied for the U.K. metric g and the Lee form w and f = exp(h). The Lee form of the metric fg is w+dh, clearly harmonic. Let the subscript f indicate a geometric object associated to the metric f g. Let SD be the Ricci tensor of the Weyl connection. The

34

CHAPTER 4.

GENERALIZED HOPF MANIFOLDS

Ricci tensor Sf of fg is related to SD by exp(ah)SD = Sf + L, where L is a symmetric (0, 2)-tensor field satisfying

L(Bf, Bf) = 0. Cf. S. Kobayashi & K. Nomizu, [162], vol. II. As SD is nonnegative we may integrate in (4.1) to yield

fMSf(Bf,Bf)*1>0. Here * is the Hodge operator of (M, g). Next we apply the Bochner formula (cf. A. Besse, [23]) to wf and obtain

(Ofwf,wf) = g(ViBf,VfB!)+JMSf(Bf,Bf) * 1. As A fw f = 0 we get V f B f = 0 and the proof is complete.

Proof of Lemma 4.1 (a sketch). The imposed condition is

Oh - m 2 2g(dh, w + dh) + d`w = 0.

(4.2)

For m = 2 this reduces to Oh = -d`w and the solution is furnished by the Hodge decomposition theorem applied to the function -d'w. For m # 2 the substitution

h=2 2log0 linearizes equation (4.2). It becomes

&0 -

m2

m2 2Od'w = 0. 2g(dO,w) +

Thus we have to look at the linear elliptic operator

L=A-m2

2i,d+m2 2(d`w)J

The problem is to find a positive element in the kernel of the formal adjoint of L:

L'=A+m2 2i,,d.

4.

GENERALIZED HOPF MANIFOLDS

35

The kernel of L' is one-dimensional and its index is 0 because L and A have the same principal part and A is self-adjoint. Let 0 be a generator of Ker(L*). Then 0 # 0. We must show that either 0 or -0 is nonnegative on M. Should the converse be true, we would find a positive function 0+ E

C' (M) so that (0, 0+) = 0, where (, ) is the L2 inner product on (M, g). Yet the orthogonal complement of Ker(L*) in COO (M) is precisely Im(L). As L has no term of degree zero, by E. Hopf's principle q5+ would be constant. Thus the generator of Ker(L*) is nonnegative. Finally, Lemma 6 of T. Aubin, [7], implies that 0 is everywhere positive.

Remark 4.1 Examples of l.c.K. manifolds which do not admit any g.H. structure may be produced as follows (cf. F. Tricerri, [258]). Let SM be the blow-up at a point of a Inoue surface SM endowed with the Tricerri metric. Then SM admits a l.c.K. metric with nonparallel Lee form. Should there exist another l.c.K. metric h with parallel Lee form w, then IIw[I would be constant. Then either w is nonzero everywhere, and then the Euler-Poincare characteristic of SM vanishes in contradiction with X(SM) = 1, or w = 0 and then h is a global Kahler metric in contradiction with bl (SM) = 1.

Remark 4.2 Vaisman manifolds cannot be Einstein, unless they are Kahlerian. This has been proved by T. Kashiwada, [147], by observing that, on the one hand, on a l.c.K. Einstein manifold the tensor field

P=

2W

N)

is hybrid, i.e. P(X, Y) = P(JX, JY) for any X, Y E X(M). And, on the other hand, a Vaisman manifold with hybrid P is Kahlerian. The natural analogue within conformal geometry of the Einstein condition from Riemannian geometry is that the symmetrized Ricci tensor of the Weyl connection be a multiple with, usually, a nonconstant factor of a chosen metric in the given

conformal structure. A Weyl structure satisfying this condition is called Einstein-Weyl. By a result of H. Pedersen & Y.S. Poon & A. Swann, [226], any compact Hermite-Einstein-Weyl manifold of real dimension greater than 6 is a Vaisman manifold. The converse is not true. The following result gives a sufficient condition for the Lee form of a compact 1.c.K. manifold to be parallel (cf. S. I. Goldberg & I. Vaisman, [115]):

Theorem 4.2 Let (M2", J, g) be a compact 1. c. K. manifold with Lee form everywhere nonzero and not exact. If the Ricci tensor of (M2n, g) is positive semi-definite and vanishes only in the direction of B then w is parallel.

CHAPTER 4.

36

GENERALIZED HOPF MANIFOLDS

Proof. Let w = df + h be the Hodge decomposition of w. Here f E C°°(M) and h is a harmonic 1-form. By a result of Bochner (cf. [23]), as the Ricci tensor is positive semi-definite, h is covariant constant. We thus have Vw = Vdf or, equivalently

V(B-Of)=0.

(4.3)

Using (4.3) we have

9(VXVyZ,B-Of) = X(9(VyZ,B-Vf)) = X(Y(9(Z,B-of)) Thus

g(R(X, Y)Z, B - V f) = 0. Set Y = Z = Et, where {E;} is an orthonormal frame and sum over i. We get

S(B - Vf, X) =0. Now the hypothesis implies

B - Vf = AB

,

AEC°O(M).

We may distinguish two cases:

a) A = 1. Then V f = 0. Thus df = 0 and w is parallel. b) A # 1. This actually does not occur. Indeed, as (1 - A)B = V f there is a function u E CO0 (M) with w = pdf . Differentiating, we get du n df = 0. As w # 0 everywhere, df is everywhere nonzero so that dp = pdf. As above

we may show that p depends upon f alone. Consequently w is exact, a contradiction.

The next result provides a criterion for a compact complex manifold to fail to admit a g.H. structure. We refer to S. Kobayashi, [159J, for the notion of a hyperbolic manifold and recall that the Hopf and Inoue surfaces are not hyperbolic. The following statement partially generalizes this state of affairs:

Proposition 4.1 A compact g.H. manifold is not hyperbolic. Proof. By a result of S. Kobayashi, [1591, a compact hyperbolic manifold has a finite group of holomorphic transformations. Thus its Lie algebra is null. But on a g.H. manifold B and JB are nonzero almost analytic vector fields.

We recall that a compact Hermitian manifold with negative holomorphic sectional curvature is hyperbolic. Thus we have:

4.

GENERALIZED HOPF MANIFOLDS

37

Corollary 4.1 A compact complex manifold which admits a Hermitian metric of negative holomorphic sectional curvature does not admit a g.H. manifold structure. In the following, we suppose w # 0 everywhere. L.c.K. manifolds satisfying this assumption are referred to as strongly non Kahler. The complex Hopf manifold carrying the Boothby metric is strongly non Kahler. If w is parallel then IIwII = 2c for some c E R, c 54 0. We adopt the notations

u=Ik'II

U=up, v=-uoJ, V= -JU.

To proceed with the study of the geometry of g.H. manifolds we collect a few technical facts in the following:

Proposition 4.2 (Cf. [269]) Let (M2s,J,g) be a l.c.K. manifold. The following statements are equivalent:

i) M is a Vaisman manifold. ii) c = IIwII/2 is constant and any of the following identities holds:

Vu=0, VU=O, V X=[U,X] Du = c{2u ®u - g}

,

DU = -cI

(4.4)

DUX = [U, X] - cX.

iii) c = IIwII/2 is constant and U is a Killing vector field for the metric g. Proof. The equivalence of i) and ii) is straightforward from definitions. The equivalence of i) and iii) follows from the identity

2g(VxU,Y) = du(X,Y) + (Lug)(X,Y), where G denotes the Lie derivative.

Proposition 4.3 (Cf. [2691) On every Vaisman manifold M the following relations hold:

LUJ=0,GVJ=0,GVg=0 [U,V]=0,dv=2c(Sl+uAv).

(4.5)

CHAPTER 4.

38

GENERALIZED HOPF MANIFOLDS

Proof. By the definition of Lie derivative, one has

(LuJ)X = [U, JX] - J[U, X] = (DuJ)X = 0. On the other hand, as J is integrable

(LvJ)X = [V,JX] - J[V,X] = [JX, JU]+ J[JU, XJ = J[JX, U] + [X, U] = J ([JX, U] - J[X, U]) = -J(,CuJ)x = 0. Furthermore, on any l.c.K. manifold we have CvSl = 0, so that (,Cvg) (X, Y) = g(JX, (GvJ)Y),

hence V is Killing. To prove the last identity in Proposition 4.3 note first

that

dv(X,Y) = 2{-X(u(JY))+Y(u(JX))+u(J[X,Y])}.

(4.6)

Then, by taking into account Proposition 4.2, we may use the identity

X(uY) = u(DXY) + c{2u(X)u(Y) - g(X,Y)}

to replace the terms X(u(JY)) and Y(u(JX)) in (4.6). This procedure furnishes

dv(X, Y) = 2c(u A v + 1)(X, Y) + 2 {(DXJ)Y - (DyJ)X }. Q.E.D.

Let (M2", J, g) be an almost Hermitian manifold. Then M is a locally conformal almost Kahler (1.c.a.K.) manifold if there is an open cover {Ui}iEI

of M and a family {fi}iEJ of C°O functions fi : Ui -+ R so that each local almost Hermitian metric gi = exp(- fi)g is almost Kahler (i.e. the 2-forms Ili given by (1.2) are closed). A 1.c.a.K. manifold with [J, J] = 0 is a l.c.K. manifold. As with 1.c.K. manifolds, on each l.c.a.K. manifold one has the Lee form w, by slightly generalizing Theorem 1.1, and, if IIw[I 0 0 everywhere on M, the unit Lee field U = IIw11-1we. A l.c.a.K. manifold is a P-manifold if it obeys the following axioms: a) IIwII = const., b) U is Killing (i.e. Lug = 0)

and almost analytic (i.e. CUJ = 0), and c) the Weyl commutant of J, given by W(X, Y) = (DyJ)X - (DXJ)Y, is orthogonal to U. Here D is the Weyl connection of M given, as in the case of a l.c.K. manifold, by

4.

GENERALIZED HOPF MANIFOLDS

39

(1.5). A P-manifold is a Po-manifold if its local almost Killer metrics are flat. Clearly, any Vaisman manifold is a P-manifold, and a P-manifold is a Vaisman manifold if [J, J] = 0. Note that all identities (4.4)-(4.5) hold on P-manifolds, as well. Finally, on any P-manifold let f be the (1,1)-tensor field defined by

f =J+v®U-u®V.

It is easy to check that

(4.7)

f3+f=0

fU=fV=0,uof=vof=0 f2=-I+u®U+v®V g(fX, fY) = g(X,Y) - u(X)u(Y) - v(X)v(Y), that is, f is f-structure with complemented frames, cf. [30]. One may obtain g.H. manifolds as appropriate circle bundles over cosymplectic manifolds (cf. Section 14.5 in Chapter 14 of this book for definitions and further references), cf. J.C. Marrero & E. Padron, [181]. An example of cosymplectic manifold may be built as follows. Let (P, J', y') be a Killer manifold. Then

,p=J'o(dpi),e= -,1]=p2dt y = pig' + p2(dt ®dt) where pi are the natural projections of P x S1 on its factors, is a cosymplectic

structure on P x S1 with fundamental 2-form '1 = p1S2', where 11' is the Killer 2-form of P. We may state Theorem 4.3 (Cf. [181]) Let (N, cp, , 77, -y) be a cosymplectic manifold with integral cohomology class [4i] and let 7r : M - N be the principle S'-bundle corresponding to [4i]. Then M is a Vaisman manifold.

Proof. There is a connection 1-form 8 in the S'-bundle it

: M N with

curvature 2-form 7r'4). Let OH denote the horizontal lift with respect to 0 of the tensor field -0 on N. Let E be the infinitesimal generator of the S1-action

on M. Set

J=cpH+(7r`rl)®E-B®CH

g=ir y+8®0. Then (J, g) is an almost Hermitian structure on M and 7r is a Riemannian submersion of (M, g) onto (N, h). Thus (cf. [2141) h[XH,YH] = [X,Y]H and

40

CHAPTER 4.

GENERALIZED HOPF MANIFOLDS

[E, XHI is vertical. Here h is the projection on the horiozontal distribution of the Riemannian submersion it. As dO = a'4i one has O([E, XH]) = 0, hence [E, XH] = 0. By the properties of cosymplectic structures and the definition

of J one may show that J has vanishing Nijenhuis torsion. Furthermore, the Kahler 2-form SZ of (M, J, g) is expressed by SZ = 7r*4? + 0 A (ir't7). Thus dfz = (ir*77) A S2, i.e. (M, J, g) is a U.K. manifold with Lee form w = 7r'77

(and Lee field B = cH). As

is parallel, one has h(OXHB) = 0. Then

9(VXHB,E) = 9(DEB,XH) = -(do)(XH,E'H) = 0 9(DEB, E) = 0. Hence OXriB = DEB = 0, i.e., B is parallel, Q.E.D. Several examples of g.H. manifolds are built in [181] by using Theorem 4.3 and the examples of compact cosymplectic solvmanifolds discovered in [183].

Chapter 5

Distributions on a g.H. manifold One of the purposes of this chapter is to study of some naturally occurring distributions on a 1.c.K. manifold, which exhibit particular geometric properties when the Lee form is parallel. Let M be a g.H. manifold. Let Dl, respectively D2, be the 1-dimensional distribution spanned by the Lee field U, respectively by the anti-Lee field V. Let V3 be the orthogonal complement of Dl ® D2. Note that D2 ® 7)3

is given by the Pfaffian equation u = 0. Since u is closed, it follows that V2 ® V3 is integrable, thus giving rise to a codimension one foliation .7 of M, referred hereafter as the canonical foliation of M. Let S be a leaf of Jro and hs its second fundamental form as a submanifold in (M, g). Recall (cf. [52]) the Gauss formula

VXY = VXY + hs(X,Y) for any X, Y E T (S), where VS denotes the induced connection on S. Yet, since u is parallel, one has u(V Y) = 0, hence VXY is tangent to S. Thus

hs = 0, i.e. S is totally geodesic in (M,g). To understand the geometric structure of the leaves of Fo we recall that given a real (2n - 1)-dimensional

CI differentiable manifold N and c E R, c # 0, a c-Sasakian structure on N is a synthetic object (gyp, £, 77, -y) consisting of a (1,1)-tensor field p, a

vector field t; E X(N), a 1-form rl, and a Riemannian metric ry, satisfying the following identities:

V =-I+7®f 77-W=0, i(f)=1

42

CHAPTER 5. DISTRIBUTIONS ON A g.H. MANIFOLD coY) = 7(X, Y) - i(X )77(Y)

(5.1)

[cp, cp] + 2(d-q) ®e = 0

dry = co,

where the 2-form 0 is given by O(X, Y) = ry(X, WY). A (2n -1)-dimensional

manifold N carrying a c-Sasakian structure is a c-Sasakian manifold. Of course, one may always go back to a usual Sasakian structure (in the sense of [25]) by a transformation: 1

1C,n=cq,ry" =c2-t.

cP=cp,

By a result of S. Tanno (cf. Proposition 4.1 in [251]) two (2n-1)-dimensional simply connected complete c-Sasakian manifolds of constant cp-sectional cur-

vature k are almost contact isometric to each other. Next, we give some examples of such manifolds. Let S2n-1 C C" be the unit sphere and let a > 0, c E R, c # 0. Let (J, g) be the flat Kahler structure of C" and set

cp=J - A®N

= - a JN, 77 = --A a , a2-a T2Y

where ry' is the metric induced on

C2

S2n-' by

(5.2)

11®171

g, N = x'8/8x' + y'8/8y' is the

unit normal on S2n'1, and .\ is the 1-form given by ,\ _ E 1(y'dx' - xidyi) , yn) are the canonical coordinates on R2n. Then , xn, y1, Also (x1, (cp, f, , ry) makes S2n-1 into a c-Sasakian manifold of constant cp-sectional curvature kc2 with k = 4/a - 3, cf. [251], [253]. Let S2n-1(c,k) denote S2n-1 thought of as endowed with this c-Sasakian structure. Next, let (M, J, g) be a Kahlerian manifold of constant holomorphic sectional curvature a whose Kahler 2-form 11 satisfies dw = cQt for some 1-form

wonMandsomecER,co0. SetN=RxMandletlr:N-4Mbethe natural projection. Set cp=

Jo(dir)-7r'(woJ)®. 8t , ° 'Y=9+17®17.

dt + 1r'w

(5.3)

5. DISTRIBUTIONS ON A g. H. MANIFOLD

43

Then (,p, l;, 77, ry) makes N into a c-Sasakian manifold of constant cp-sectional

curvature k = £-3c2 (cf. [177], [208]). Let (RxM)(c,k) denote the manifold N thought of as endowed with this c-Sasakian structure. In particular, if C"(0) is C"-' carrying the flat Ki hler metric bjjdz'®dz', then (I = 0 and) (R x Cn-1(0))(c, -3c2) = R2"-1(c) is a c-Sasakian manifold of constant cpsectional curvature -3c2. Similarly, if B`(I) is the open unit ball in Cn-1 of constant holomorphic sectional curvature I < 0 (cf. [162], vol. II, p. 169) then (R x Bn-1)(c, k) is a c-Sasakian manifold of constant (p-sectional curvature k = f - 3c2. Going back to the geometry of g.H. manifolds, we may state:

Proposition 5.1 (Cf. [269]) Let M be a Vaisman manifold. Let S be a leaf of .Pp and i : S M the inclusion. Let (cp, , 77, y) on S be given by

cp= Jo(di)+(i'v)®(Uoi)

f =V oi, r/=i*v

(5.4)

ry = i'g Then (gyp, l;, n, y) is a c-Sasakian structure on S. If in addition M has flat local Kdhler metrics then (S, y) is an elliptic space form of sectional curvature c2, where c = IIwII/2

Proof. It is an easy matter to check that (cc, l;, rl, y) satisfies Properties (5.1), hence it is c-Sasakian. Assume now that D is flat. Then (1.5) gives

R(X,Y)Z

= c2{[u(x)Y - u(Y)X]u(Z) + [g(X, Z)u(Y) - g(Y, Z)u(X)]B + g(Y, Z)X - g(X, Z)Y}

On the other hand, we may use the Gauss equation of S in M (cf. [52]):

g(R(X,Y)Z,W)

=

y(RS(X,Y)Z,W) g(hs(X, Z), hs(Y, W)) + g(hs(Y, Z), hs(X, W)),

for any X, Y, Z, W E T (S), to get

RS(X, Y)Z = c2{y(Y,Z)X - y(X,Z)Y}. Q.E.D.

Using

g=y+u®u and the De Rham decomposition theorem, we obtain

(5.5)

44

CHAPTER 5. DISTRIBUTIONS ON A g.H. MANIFOLD

Proposition 5.2 (Cf. [269]) The universal Riemannian covering manifold M of a complete Vaisman manifold is the Riemannian product of a simply connected c-Sasakian manifold N, which is the universal covering space of a leaf N of 7o and the real line.

Let us look at the distribution DI ® D2. It is involutive (as [U, V] = 0) thus giving rise to a dimension 2 foliation of M. We collect its properties in the following (cf. [275] and [278]):

Theorem 5.1 On any Vaisman manifold M the distribution Dl ®D2 is a complex analytic foliation whose leaves are parallelizable complex analytic manifolds of complex dimension 1. The leaves are totally geodesic, locally Euclidean submanifolds of M, the foliation is Riemannian, and in any foli-

ated chart the metric splits as

g = gay(z,z)dza ®dz + (u + iv) ®(u - iv).

(5.6)

The first term in the decomposition (5.6) is a Kahler metric whose fundamental form is -dv. Proof. The properties of the leaves follow from the identities

VUU=VUV =VvU=VvV =0 showing that the leaves are complex analytic and totally geodesic and from

Lug =£vg=0, proving that g is bundle-like. By the classical Frobenius theorem there exists a local coordinate system with respect to which D1 ® D2 is (locally) given by za = const. , a E { 1, , n - 11. This yields (5.6). Finally, the last statement in the theorem follows from Proposition 4.3. See also [64].

Remark 5.1 A comparison between the decompositions (5.5) and (5.6) leads to the explicit formula for y, that is -y = 2g_(z, z)dza ®dz + v ®v.

(5.7)

Compare with D.E. Blair, [25] (for the canonical form of a Sasaldan metric). In particular, this provides another proof of Proposition 5.2. In particular, if M is a compact Vaisman manifold, the existence of the complex analytic vector field U - V1_-_1V leads to the following (cf. I. Vaisman, [274] ):

5. DISTRIBUTIONS ON A g.H. MANIFOLD

45

Corollary 5.1 (Cf. [275]) All the Chern numbers of a compact g.H. manifold vanish. In particular, the Euler-Poincare characteristic of a g.H. surface vanishes.

The next result (cf. K. Tsukada, [264]), important in its own right, provides a large class of complex compact manifolds not admitting any locally conformal Kiihler metric:

Theorem 5.2 On a compact generalized Hopf manifold, the foliation Dl D2 has at least one compact leaf. Proof. Let i be the Lie algebra of infinitesimal holomorphic isometries (i.e.

vector fields X satisfying LXg = 0, LxJ = 0; it contains U and V). Let c be its Lie subalgebra generated by U and V. c is abelian because [U, V] = 0. Let C be a connected Lie subgroup of Auto(M), the connected component of the Lie group of holomorphic isometries of M) corresponding to c. Then the leaves of Dl ®D2 are orbits of C. Let T be the closure of C in Auto(M) and t its Lie algebra. T is a toroidal subgroup of Auto(M). As every C-orbit is contained in a T-orbit, we have to show that there is at least one T-orbit which is also a C-orbit. For a X E t, let Zerou(X) = (p E M;7rq(Xp) = 0}, where i3 denotes the

orthogonal projection of TM on D3. It is easily seen that Zerou(X) # 00 and is T-invariant. Hence, each of its connected components is a compact generalized Hopf manifold invariant at the action of T. One can now show inductively the existence of a compact T-invariant submanifold N of M on which 7r3(X) = 0 for each X E t. This is the requested T-orbit, the compact leaf we were looking for.

Now we can prove the announced non-existence result:

Theorem 5.3 (Cf. [2641) The product of two compact generalized Hopf manifolds does not admit any locally conformal Kahler metric. Proof. Let M1, M2 be the two compact g.H. manifolds and T1, T2 their compact leaves (one-dimensional complex tori) provided by the above theorem. Let pl, p2 be the canonical projections of M = M1 X M2 on M1, M2, i1, i2 the canonical inclusions of M1, M2 in the product and j1, j2 the inclusions

ofT,rT2inM,,M2. By absurdity, M, x M2 admits a I.c.K. metric g with Lee form w. This implies the existence of some closed 1-forms wl, w2 on Ml, M2 such that [w]=[plwl] + [p2w2] in Hl(M, R). Making, if necessary, a conformal change of g, we may suppose w = piwl + p2 w2. g induces a 1.c.K. metric iig on

46

CHAPTER 5. DISTRIBUTIONS ON A g.H. MANIFOLD

M1 with Lee form wl = iiw. Now, by a result we shall prove in Chapter 7, (namely by Theorem 7.12), we see that [jlwl] 0 in H1(T1,R). Similarly, W2w2] # 0 in H1(T2,R). Let j be the inclusion of the compact complex submanifold T1 x T2 in M. Then the induced metric j*g is an l.c.K. metric on T1 x T2 with Lee form j*w = ji wl + j2w2. Hence [j*w] # 0 in H1 (Ti x T2, R) i.e. j*g is not globally conformal Kiihler. On the other hand, T1 x T2 bears also a Kahler metric, the product one. Therefore, by Theorem 2.1 any l.c.K. metric on it must be g.c.K. This contradiction completes the proof.

Note that although, in general, the product of two l.c.K. metrics is not l.c.K., this result says a lot more. Remark 5.2 If the foliation Dl ®1)2 is regular, then Theorem 5.1 says that its leaves are complex curves on M. This provides examples of compact complex manifolds which admit no g.H. manifold structure with a regular foliation D1 ® V2 with compact leaves. Indeed, the Inoue surfaces Sly are known to possess no compact complex curves (cf. M. Inoue, [141]) and the Hopf surfaces Ha,p (given by H«.a = (C2 - {(0,0)})/Ga,Q, where G,,,0 is the discrete group generated by (z, w) '-+ (az, /3w)) which (by Proposition 18.2 of W. Barth & C. Peters & A. Van De Ven, [13], p. 173) admit only two compact complex curves (the projections of the coordinate

axis in C2), provided that all

3t for any (k,e) E (Z X Z) - {(0,0)}.

Summing up, we may state at least:

Proposition 5.3 The Inoue surfaces Sly and the Hopf surfaces H0,p admit no g.H. metric with a regular foliation D1 ® D2 with compact leaves.

We end this chapter by looking at V. Unlike the previously discussed distributions, V3 is not integrable. Precisely, we may state the following:

Proposition 5.4 Let N be an integral manifold of V3. Then:

i) dimRN 0: ptu(x) = x} where ptU is the 1-parameter group of transformations generated by U. As U is Killing, its orbits are geodesics. Let then y be an orbit through x and y' another orbit, sufficiently close to -y so that there is a minimal geodesic

beginning at x and meeting y' orthogonally at a point x'. As ptu is an isometry, it maps the arc xx' onto another arc orthogonal to y and y', too. Hence, when x moves along y for a whole period, so does the corresponding point on y'. This yields LU = const. We may assume (up to a homothety) that LU = 1 and C = S1. Moreover, it is easy to check that the action of S1 on M (as the 1-parameter group of U) is effective and free. To organize M as a S'-principal bundle, let {Wa} be an open cover of M consisting of cubic neighborhoods obtained from the regularity of D1. Then {7r(Wa)} is an open cover of N. Define the local sections Sa (x 1 ,

where x1, ,

,

.

.

2n-1) = (x1 , . )x

,x2n-1, Ct.)

x2n-1 are the coordinates on 7r(Wa). Now the functions

Ft.: 7r (Wa) X S1 - M , fta(y, t) = PtU(sa(y)) provide the desired fibration. Furthermore, note that u may be identified with a connection 1-form in this fibration. Indeed GUu = 0 hence u is Slinvariant. Next, let A = d/dt be the basis of the Lie algebra s = R of S'

and set u = u ® A. If B E s then let b be the fundamental vector field induced on M. For u to be a connection form we need that u(B) = B and that u is S1-invariant. Yet A = U so that u(A) = A. The right translation with t E S1 is precisely ptU which clearly leaves u invariant. On the other hand, since du = 0, the connection u is flat.

Let S be a leaf of Dl. Then Iris : S - N is a local diffeomorphism and hence a covering map. Thus S and N have the same local differential geometry. In particular N admits a natural Sasakian structure. N2n-1 be a flat principal Sl-bundle over a Conversely, let 7r : Men Sasakian manifold with structure (cp, , rt, y). Let U be the vector field on M whose trajectories are the fibres of 7r and let u be a flat connection 1-form (i.e. du = 0, u(U) = 1, GUu = 0). We endow M with the metric g given by

g=7r*y+u®u.

(6.1)

Then U is orthogonal (with respect to g) to the horizontal distribution of the fibration. As u is closed, the horizontal distribution is integrable thus

6.1. REGULAR VAISMAN MANIFOLDS

51

giving rise to a codimension 1 foliation of M. Moreover g(U, U) = 1 and a decomposition of X E X(M) as X = X' + AU yields g(X, U) = A. Let 1) be the fundamental 2-form of N (i.e. 4 (A, B) = 'y(A, cpB)). Define (on M) the 1-form v and the 2-forms 41, S2 by setting

v=-9r`77, 41 =7r'4) , S2='I'+uAv.

(6.2)

It is immediate that rank(S2) = 2n. We define a (1, 1)-tensor field J on M by setting g(X, JY) = S2(X, Y). (6.3)

Let us check that J is an almost complex structure. To this end, note that

ry(7r.X,ir.JY) + u(X)u(JY) = -v(X)u(Y) + v(Y)u(X) +-y(ir.X,Wir.Y) u(JY) = Sl(U,Y) = v(Y) = -ij(7r.Y) .y(ir.X, 7r.JY - cp7r.Y) = -i(7r.X)u(Y).

In the last identity, let us replace ir. X by cpX', X' E X (N). We obtain 'y(cpX', ir.JY - cp7r.Y) = -'y(X', cp7r.JY - cp27r.Y) = 0

As X' was arbitrarily chosen, the last identity yields

co r.JY = cp2ir.Y = -7r.Y + ii(ir.Y) = -7r.Y By (6.2) we get

g(X, J2Y) = S2(X, JY) = -g(X,Y)

from which j2 = -I. Furthermore g(JX, JY) = f2(JX,Y) - Sl(Y, JX) = g(X, Y)

so that (M2", j, g) is an almost Hermitian manifold with the Kahler form n. Next, from di1 = 1), as N is already Sasakian, we derive dv=%F

,

d1I=uASl.

It remains to be shown that J is integrable and U Killing. The integrability of J follows from the normality condition on N. As to the vector field U, note first that the space of its trajectories may be identified with N. Thus

M is a regular manifold. We take an open cover with cubic coordinate

CHAPTER 6. STRUCTURE THEOREMS

52

neighborhoods. Let these coordinates be (x=, t) with U = 8/8t. Using these coordinates, the identity (6.1), and GUU = 0, one may show that U is Killing.

An application of the previous result, within Kaluza-Klein theory, was given in S. Ianu§ & M. Visinescu, [139]. A similar result may be established for the foliation D' ®D2. Precisely, we have (cf. [275] and [278]):

Theorem 6.3 Let (M2n, J, g) be a compact, connected, Vaisman manifold whose foliation D1 ® D2 is regular. Then the projection

p:M-+P=MI(D1(D D2) is a TC1 -principal fibre bundle over a compact Hodge manifold. Moreover u - iv is a connection form and its curvature projects (up to some constant factor) onto the Kahler form of the base manifold.

Proof. Let us denote by LU, LV the periods of the vector fields U, V. As above, one may show that LU, LV are finite constants (real numbers). Now let r be the group generated over Z by LU and iLV, where i = r--1. As LU + iLV is constant on M, we may define a free analytic right action of C/r TC on M by setting z(x) = ptu e psv(x) , z = t + is.

(6.4)

On the other hand, let {Wn } be an open cover of M. It is convenient to consider complex coordinates (zl, . , zn) adapted to the foliation D1ED D2. Let Wa = p(W,,) C P be the corresponding coordinate neighborhoods on P. We define local sections Sa(z 1

. .

. ,

zn-1)

= (z1,. ..,zn-1,0)

and trivialization homeomorphisms hua : Wa X TC1 - P -I (W.,

by setting 1

n-1

1

n-1

where r E TC1 and the action is described by (6.4). This ends the proof of the existence of the principal Tc-bundle structure. As to the differential form cp = u-iv, it is enough to show that it vanishes on the horizontal distribution and that U and V are infinitesimal automorphisms of cp. The curvature of the connection form cp is dcp+ [cp, cp]/2 = -idv.

6.1. REGULAR VAISMAN MANIFOLDS

53

We endow P with the metric induced by the first term in the decomposition (5.6). Its Kahler form is -dv so that P is a Kiihlerian manifold. To show that P is actually a Hodge manifold, we look at the Sasakian manifold N constructed in the previous theorem. By a classical result of W.M. Boothby & H.C. Wang (cf. [37] and our Appendix A) the orbit space N/ is a compact Kahler manifold, the base manifold of a principal S'-bundle of total space N, and -Gvv is a connection form. As is the projection of V, it is easily seen that the base spaces of the two bundles coincide. Consequently -/.vv represents an integer cohomology class.

Remark 6.1 In the previous construction the fibration is never trivial because V3 is never integrable. Moreover, the above construction is not reversible, in the sense that not every principal Tc-bundle over a compact Hodge manifold carries a g.H. metric. Indeed, we may view a CalabiEckmann manifold as the total space of a torus principal bundle over a product of two complex projective spaces by simply considering the direct product of the appropriate Hopf fibrations. Yet the Calabi-Eckmann manifold admits no l.c.K. structure because, by the simple connectedness the l.c.K. structure would be g.c.K., and the Calabi-Eckmann manifolds are known to admit no global Kahler metrics, for topological reasons. More generally, by a result in A. Blanchard, [33], the total space X of a principal T"-bundle over a Kahler manifold P is itself Kahler if and only if b = 0, where b is the morphism in the exact sequence:

0-H'(P,C)-,H'(X,C)-,H'(T,C)-H2(P,C) Conditions for the total space of such a principal bundles to be simply connected are given in T. Hofer, [130], while Theorem 3.5. of I. Vaisman, [275], determines all Tc-bundles whose total space is a regular g.H. manifold.

Remark 6.2 (Cf. [275]) From the last part of the above proof one gets a commutative diagram of principal fibre bundles associated with a compact strongly regular g.H. manifold. Namely

Tc

M

-+ M p

1

1

S' it

N , S'

P q

where q denotes the Boothby- Wang fibration. The fact that p factors in two S'-fibration may be seen as follows. Let U be a trivialization neighborhood

CHAPTER 6. STRUCTURE THEOREMS

54

for q. Then Nju Nju. Now

U x S' and MpU is a flat principal circle bundle over

H2(NIU, Z) = H2(U x S1, Z) = H2(Sl, Z) = 0

so that MjU is trivial over Nju and the proof is complete. On the other hand, applying twice the Gysin sequence for the fibrations p and q, and using the vanishing of X(M) we get the following relations satisfied by the Betti numbers of M and P

b; (M) = bi(P) + bi-1(P) - bi-2(P) - b;-3(P) , 0 < i < n - 1 bn(M) = 2 (bi-1(P) - bn-3(P)) , i > n + 1. In particular, when M is a compact homogeneous g.H. manifold, the base Hodge manifold is simply connected and has vanishing odd Betti numbers (cf. A. Diaz-Miranda f4 A. Reventos, (1921). Thus a compact homogeneous g. H. manifold has b1(M) = 1. The following converse of Theorem 6.3 is also available:

Theorem 6.4 (Cf. [264]) Let 7r: M N a holomorphic Tom-bundle over a compact Kahler manifold. If there exists in this bundle a connection form z(i with the properties

(i)zboJ=Vr--1z/i, (ii) there exists a Kahler form Sl' on P such that the curvature form 91 of ip satisfies T = c7r*SZ' for a nonzero real constant C, then M has a generalized Hopf metric whose leaves of D1 ® D2 coincide with the fibres of the bundle.

Proof. Let' = 0 + mow. By (i) we know that 0 = w o J and w = -o o J. The curvature form is given by T = dpi = do + V1_-_1dw. Then, by (ii), dw = 0 and dO = cir*Sl'. We let g' be the Kahler metric on P corresponding to W. Now we can define a metric on M, setting

9-7*g

+21®3;

*9'+w®w+o®o.

One checks easily that g is Hermitian and invariant to the action of the torus. Its fundamental two-form will be Sl = 7r*S1' + w A 0, with derivative dSl = (cw) A fl. Hence g is a l.c.K. metric. To show that w is parallel, identify

C with the Lie algebra of T. The fundamental field a* corresponding to A E C is an infinitesimal isometry of g. Let B be the fundamental vector

6.1. REGULAR VAISMAN MANIFOLDS

55

field corresponding to a = v1--1. Then B = wd and, being an infinitesimal isometry of g, from Proposition 4.2 we conclude g is a generalized Hopf metric. Following K. Tsukada, [264], we indicate a way to construct holomorphic toroidal bundles over compact Kahler manifolds imitating the construction of Hopf manifolds as tori bundles over complex projective spaces. Namely, start with a compact Hodge manifold P and with a negative holomorphic line bundle E. Let 7r : F - P be the corresponding principal C*-bundle

(i.e. F = E - { zero section }). Now fix a E C*, I a 1< 1 and let I'a be the cyclic subgroup of C* generated by A. Finally define M as the quotient

complex manifold F/I', Choose a Hermitian fibred metric on E. It has a Hermitian connection with connection form 0 satisfying ip o J = Vr--1 0. Let IF be its curvature form. This is a basic form, so there exists a real, closed 2-form y of type (1, 1) on P such that 7r*y = -(27rvr--1)-1T. y represents

the Chern class c(E) in H2(P; C). Since E is negative, we may assume y negative definite in each point of P. Now let a = (21rv'-1)-1 log A. The imaginary part of a is positive because A has subunitary module. Let A be the lattice generated by 1 and a and set T = C/A. Letting p : C -+ T be the canonical projection, we define a homomorphism p : C* -+ T by log f ). It induces the Lie algebra homomorphism dp acting as dp(z) = (21rv '-1)-1z, z E C. Observe that T is isomorphic as a complex Lie group with C*/I'a as kerp = r.\. This proves that M = F/I'a is a holomorphic T-bundle and the natural projection p : F -+ M is a bundle homomorphism with corresponding homomorphism

p. Let 0' be the unique connection form on M corresponding to '0 via A. If 'P' is its curvature form, then p*Vi' = dpV; and p*W' = dpW (cf. [162], chapter II). Finally, if 7r' : M -+ P is the projection, the above equalities

imply 0' o J = / T ' and W' _ -7r'*y. Since y is negative definite, the connection form "" satisfies the conditions of the theorem. For the smaller class of Hermite-Einstein-Weyl manifolds (which, as previously indicated, in real dimension greater than 6 are g.H. manifolds) the

last theorem admits a natural sharpening. Indeed, using the same notation, if M is Einstein-Weyl then P is a Kahler-Einstein manifold of positive sectional curvature. Thus, by a result of S. Kobayashi, [158], P is simply

connected. To achieve the inverse construction, note that in this case N must be the circle bundle associated to a rational power of the canonical bundle. On the other hand, N is Einstein, too, and has positive scalar curvature, hence it has a finite fundamental group. Hence we may, by passing to the universal cover, assume that N is simply connected and that M is

56

CHAPTER 6. STRUCTURE THEOREMS

a trivial circle bundle over it. Now we see that the universal cover of M is L - {0} where L is a maximal root of the canonical bundle. In the conformal class of M we choose a metric g with parallel Lee form w. On the universal cover w = df and g' = exp(- f )g is Kahler and Ricci flat. One easily shows that B(f) = 11w112 and (JB)(f) = 0 so that C` acts on the universal cover by homotheties with respect to g'. Summing up, we have the following:

Theorem 6.5 (Cf. [226]) Each compact Hermite-Einstein-Weyl manifold M (of real dimension > 6) which is strongly regular and is not globally conformal Einstein, arises as a fibration over a compact Kahler-Einstein manifold P (of real dimension 2n-2) of positive scalar curvature. Moreover, 111 is obtained as a discrete quotient of the Ricci flat Kahler structure on a principal C`-bundle associated to a maximal root of the canonical bundle of P.

6.2

L.c.K.o manifolds

We further call locally conformal Kahler flat (briefly l.c.K.o) those 1.c.K. manifolds whose local Kahler metrics are flat. In other words, the Weyl connection is flat. In particular, the local Kahler metrics have nonnegative Ricci tensor so that, by applying Theorem 4.1, we see that a compact l.c.K.0 manifold is globally conformal with a g.H. manifold, and this yields the results we report in the remainder of this chapter. Before doing so, let us note that if dimc M > 3 then a l.c.K.o metric is merely a conformally flat l.c.K. metric. This is easily seen because a conformally flat Kahler metric is flat. On the other hand, this is not true in complex dimension 2. Indeed, it suffices to look at the product of the Poincare half plane with the projective line (carrying the product metric). Instead, the result remains true for compact l.c.K. surfaces (cf. M. Pontecorvo, [231]). The structure of compact l.c.K.0 manifolds is best understood. We begin with

Theorem 6.6 (Cf. [269]) Let (M2',J,g) be a compact l.c.K.0 manifold. Then its Riemannian universal covering manifold is C" - {0} endowed with the metric 9o = 41zl -2

E dzi ®dzj

and g is globally conformal to the metric induced by go on M. Moreover, M has the same Betti numbers as the complex Hopf manifold.

Proof. As remarked above, we may suppose the metric g with parallel Lee form. Then the covering space M is S x R where S is the covering space of

6.2.

L. C. K.o MANIFOLDS

57

a leaf S of Fo. If the local Kahler metrics are flat, then by Proposition 5.1, each leaf of T0 has constant sectional curvature c2. Moreover, any leaf S is complete as S is totally geodesic and M complete. Then S is a sphere, not necessarily of unit radius, endowed with the usual Riemannian metric. Set S = Stn-1(r). We have the diffeomorphism

NR2n-{0} (u,t) - (u,exp(t/r)) H uexp(-t/r) where E(ui)2 = r2. By this diffeomorphism, the metric F_(dui)2 + dt2 transforms into ds2

T2

E(d_-')2.

= E(xi)2

Passing to complex coordinates we find 2

ds2

= 2 zi >

dzk ®dzk

and the proof of the first assertion in Theorem 6.6 is complete. To compute the Betti numbers of M, following S.I. Goldberg, (1061, we consider, for each p-form a (with 0 < p < dim M), the quadratic form

Fp(a) =

ti2...ip

si2...ip +

kl P 2- 1 i,jktai3...ip,

where It? are the components of the Ricci tensor of g. Using (6.5) we get

Fp(a) = c2{p!(2n - p -1)IIaII + 2(p -1)!(p - 1)IItuaII},

(6.5)

where tX denotes the interior product with the vector field X. Thus F,(a)

is positive definite for n < p < 2n - 2 so for these values of p we have bp(M) = 0. From Poincare duality we have also bp(M) = 0 for 2 < p:5 n. Furthermore, it is clear that bo(M) = b2i,(M) = 1. As for b1(M) we note that as u is parallel, it is harmonic (yet not exact) so that b1(M) < 1. Now let /3 be a harmonic 1-form. Then */3 is also harmonic and Fp(*/3) = 2c2(2n - 2)!(n - 1)IItu *)311 ? 0.

One now shows that V(*/3) = 0 and cu * /3 = 0. Again applying the Hodge operator we derive u A,3 = 0 from where /3 = Au. Yet /3 is closed so that dA = Au and from 6/3 = 0 we conclude that dA is orthogonal to u. Thus any harmonic 1-form is proportional to u. This concludes the proof.

CHAPTER 6. STRUCTURE THEOREMS

58

We proceed by studying the group G of all covering transformations of a compact l.c.K.0 manifold (following I. Vaisman, [2751). The elements of G preserve the metric go. In particular they are conformal transformations of

(C'2 - {0}, E dz' 0 dzi). Moreover, they preserve the Levi-Civita connection of this metric. Thus an element of G is an affine transformation of C" which fixes the origin. Consequently, for any y E G there is p > 0 (called the module of -y) and [ajk] E U(n) so that

-y(z) = pajz To further describe G we need the following (cf. K. Kodaira, [163]):

Lemma 6.1 Let -y E G with p(-y) < 1. Then 7 generates an infinite cyclic group {y} of finite index in G. In addition, there exists an element -yo with p(yo) < 1, maximal among all the elements with subunitary module. Proof. Let -y E G with p(y) < 1. Then p(ryk) = p(-y)k are distinct real numbers so that {-y} is infinite and (C" - {0})/{y} is compact. This is a covering space for M, hence C/{-y} is finite. By the compactness of M

there is an element of subunitary module. Now, we have a finite number of equivalence classes [g] in G/{ y} and all the elements in a class have modules of the form p(g)p(A)k. Thus, in each class we may choose a maximal element with subunitary module. As there is just a finite number of classes, the result is proved. Consider the subgroup H = {ry E G : p(ry) = 1}.

(6.6)

Note that H is a normal subgroup of G. Moreover Theorem 6.7 H is a finite subgroup commuting with -yo and

G={hyo:hEH,kEZ}.

(6.7)

Proof. The second statement follows from the normality of H in G. Let [g] E Gl {yo}. If there is a representative h E H of this class, then the module of any other representative will be p(h)p'(-yo) = pk(ryo) # 1. In this case k # 0 and h is unique in [g], hence H is finite. In the general case, let (g-yo} be an equivalence class and A an element of subunitary module, maximal in this class. Then p(A7 ') > p(A) so that p(A-yo') > 1 and p(A) > p(-to). By our choice of the elements A and -yo we obtain p(A) = p(-yo)

6.2. L.C.K.0 MANIFOLDS

59

and thus A-yp 1 E H. Thus any equivalence class has a unique representative

in H. The proof is complete.

At this point, we may state the following characterization of compact l.c.K.0 manifolds (cf. I. Vaisman, [275]):

Theorem 6.8 The compact l.c.K.o manifolds are completely described by M = (Cn - {0})/G with G given by (6.6) - (6.7). Proof. We have already shown that compact 1.c.K.0 manifolds are of the announced type. Now, let H be a unitary finite group and 'yo a transformation which commutes with H and has the form ^to(z) = po exp(2iriAk)zk.

We define the group G by (6.7) and set M = (Cn - {0})/G. Then, when M is compact, it is a compact l.c.K.0 manifold. A slightly different approach is provided by K. Tsukada, [261]. Starting with the observation that the universal cover of a g.H. manifold is the product of a Sasakian manifold with R (cf. I. Vaisman, [269]) K. Tsukada computes the deck transformation group and obtains

Theorem 6.9 Let M be a compact g.H.o manifold (i.e. a l.c.K.0 manifold with parallel Lee form) whose Lee form has unit length. Then M is (S x R)/t where S is a compact Sasakian manifold of constant sectional curvature 1 and r is an infinite cyclic group generated by 1)(x, t) = (cp(x), t + b) where

cp is an automorphism of the Sasakian manifold S and b a nonzero real number. Moreover, if S = S2i-1 (1) then M is biholomorphically isometric to Ma,b = (Cn - {0})/Ga,b carrying the Boothby metric where Ga,b is the group of biholomorphic isometries exp(-21rb) diag (exp(21ria1),

b > 0, a = (al, ...,an)

,

,

exp(21ria,,))

,

-1/2 < al < ... < an < 1/2.

Proof. We first need to compute the deck transformation group f of the universal covering 7r : M = S x R - M where S is Sasakian. Of course, the Lee form on M is w = -dt. Any (D E f is a biholomorphic isometry of k hence V(-dt) = -dt and d(Vt-t) = 0. This gives Vt = t+b. On the other hand, must transform a slice S x {t} into a slice S x {t + b} preserving the Sasakian structure on S. Then 1)(x, t) = (¢(x), t + b), where 0 is an

CHAPTER 6. STRUCTURE THEOREMS

60

automorphism of the Sasakian structure of S. This gives a homomorphism a : r -+ R, a(4) = b. Its kernel acts freely and as a properly discontinuous group of transformations of S x R. Hence we have

(S x R)/Ker(a) = S x R

with S = S/Ker(a). Set r = r/Ker(a). Then M = (S x R)/r. At this point we assume S compact and determine the group F. Of course r is isomorphic to a(I'), a subgroup of R. We claim it is discrete. Otherwise there exists a sequence {(D"} of elements in r so that a(4D,a) converges to 0.

Yet 4)"(x, t) = (0"(x), t + aFor each fixed x E S pick a subsequence (x) 1, convergent to some y E S (because S is compact). This gives lim D"k (x, 0) = (Y'0) koo

in contradiction with the fact that r is a deck transformation group. Being discrete in R, a(r) is infinite cyclic and then so does I'. The last assertion of the theorem follows by noting that the automorphisms of the Sasakian structure of S2i-1 (1) are precisely the unitary transformations of C" restricted to Stn-1(1). For any A E U(n) there is U E U(n) so that U-1AU = diag(exp(27ria1),

,

exp(27ria.,i)).

The proof is complete. Remark 6.3 Examples of Vaisman manifolds which do not admit any 1.c.K.o

structure may be obtained as follows. One considers T to be the total space of the induced Hopf fibration over an irreducible projective algebraic curve of genus g. One may compute the Betti numbers of T as bo(T) = 1, b1(T) = 2g + 1, b2(T) = 4g. Therefore p9(T) = g and if g > 1 then T must be in the Kodaira class VI, while, from the previous classification, l.c.K.0 surfaces may only be in the Kodaira class VII.

6.3 A spectral characterization The aim of this section is to give a spectral characterization of the Hopf manifolds due to K. Tsukada, cf. (2611. Let 14 and Op,y be the real and complex Laplacians acting on r-forms, respectively on complex forms of type (p, 4)

6.3. A SPECTRAL CHARACTERIZATION

61

Theorem 6.10 Let CH,' (with n > 3) be a Hopf manifold carrying

the

Boothby metric so that the generator a satisfies - 27r Re log a >

2 2n + 1

If a compact connected Hermitian manifold M satisfies Spec(M, Op,q) = Spec(CHQ, Op,q)

for (p, q) E {(0,0),(0,1),(1,0)} and Spec(M, Ar) = Spec(CHQ, A,) for r E {0,1 } then M is biholomorphically isometric to CHQ.

Proof. We first prove that M is 1.c.K. This will follow from the equality of the complex spectra. Indeed, let (AP', A2'q, } be the spectrum of Op,q and 00

00

bp.gtk E k=0

j=1

the corresponding asymptotic expansion. The first coefficients bk'q were com-

puted by H. Donnelly and P. Gilkey (cf e.g. (1221). Let us denote by T the torsion of the Chern connection VC. Setting K=Jrs*1 M

K1= I T

K,

7T

we have 0o b1' =

2n-1 6

(K + 3K2)

n-1 2n-1 bi'0 0 b1,

=

12

[2(n - 3)K - 3K1 + 6nK21

2n-1

= 12

[2(n - 3)K + 9K1 + 6(n - 1)K21.

It is now clear that the equality of the complex spectra implies the equality of the invariants K, K1, K2. The statement follows from the fact that the Lee form is, modulo a multiplicative factor, the trace of the torsion of VC. Actually we prove a slightly stronger result, that is

CHAPTER 6. STRUCTURE THEOREMS

62

Let M, M' be two compact connected Hermitian manifolds of complex dimensions > 3. If

Spec(M, p,q) = Spec(M', .,q) for any (p, q) E {(0, 0), (0, 1), (1, 0)} then M is 1.c.K. if and only if M' is l.c.K.

Let w be the Lee form of M. To proceed, we first note that due to the fact that the Boothby metric of a complex Hopf manifold is conformally flat, M must be conformally flat, too. This follows from the equality of the real spectra. Indeed, one has the more general statement Let N, N' be two compact connected Riemannian manifolds of real di-

mension > 4. Let N' be conformally flat with constant scalar curvature s'.

If Spec(N, A,) = Spec(N', A,.) , r E {0, 1}

then N is conformally flat as well and has the same (constant) scalar curvature. Moreover IN IItI2 * 1

=

IN' II'II2 * 1.

Cf. also D. Perrone, [228). For the proof, let {A , a2,

of 0, and

00

E exp(-tar)3 - (47rt)-' /2 j=1

} be the spectrum

00

E aktk

k=0

the corresponding asymptotic expansion. The first coefficients are well known (cf. e.g. [22]):

a0=vol(N),a0= 6JNS a2

360 ,IN

(5s2

- 2IISII2 + 2IIRII2 * 1

ao=mvol(N),a1=m6 J $ * 1 (5s2

a

IN 12

IN

- 2IISII2 + 2IIRII2 * 1

(11R112 -6

IISI12

+ 282) * 1.

Now, as a2 = a'2 and al = a'2i using the known relation connecting the length of the curvature tensor with that of the Weyl tensor W IIWII2=IIRII2_m4

2

2IISII2+(m-1)(m-2)s2'

6.3. A SPECTRAL CHARACTERIZATION

63

we derive 13m3 - 67m2 + 100m - 36 s2" (5 (m - 2) I I W I I2 +

N

=

L' \15(m-2)IIW1II2+

(m - 1)(m - 2)

J

*1

13m3-67m2+100m-36s2

(m-1)(m-2)

As m = dim(N) > 4, the polynomial 13m3 - 67m2 + 100m - 36 is strictly positive. As s' = const., the equality of the coefficients a2, a'a, respectively a?, a'?, together with the Schwarz inequality imply IN

*s2*1. JJ'

Finally, W' = 0 implies W = 0 and s = s'. The last equality in the statement follows by using a2 = a'2 once more. So, as CH, is conformally flat and has constant scalar curvature s' = 2(n - 1)(2n - 1), so does M. We wish to show that M has a parallel Lee form of unit length. This can be done as follows. By standard formulas in conformal geometry (cf. e.g. [23J, p. 59) one gets

s = 2(2n - 1)bw + 2(n - 1)(2n - 1)IIwII2 IISII2

= 4(n - 1)2IIVwII2 + 8(n - 1)2(Viw=)w'w' + 4(n - 1)(4n - 3)IIWII26W + 4(n - 1)2(2n - 1)IIdwII4 + 2(3n - 2)(bw)2

So, from the first formula we derive (2n - 1)(1 - 11w1I2) = bw.

To exploit the relation IN

we must integrate the second of the above two formulas. We start by integration in 6(IIwII2w) = -2(Vjw;)w'w' + IIwII2bw.

This gives

[

(V3wi)ww'*1=2IMIIwII2bw*1.

CHAPTER 6. STRUCTURE THEOREMS

64

Also taking into account JM (IIw112 - (b ,)2 -11IdwII2 + S(wa,wJ)) * 1 = 0,

we get fxfIIVw112*1=(n-2)fm IIWI126W*1+ fM(5w)2 * 1.

So, on the one hand we have !CH^ 11`'112*1 = fM ISII2 * 1 a

_ -2(n-2)f (bW)2*1+ My

2n1

1

fm

s2*1

and on the other

fCH2n - 1 J (15r112

*1=

r2

1

H'

s

*1.

This yields bw = 0, hence IIw1I = 1. We have proved that M is a conformally flat generalized Hopf manifold.

By the first part of Theorem 6.6 we have M = (S x R)/G with S a Sasakian manifold of constant sectional curvature 1 while G is infinite cyclic generated by the transformation -,D(x, t) = (cp(x), t + 20b'). We shall show that S is

isomorphic to Stn-1. To this end, it is enough to check that S is simply connected. We need some preparation. Let S1 = R/(2¢b'Z). Let zr : M S' be the natural map. It is a Riemannian submersion with totally geodesic fibres. Let As be the Laplacian acting on C°D(S). With the usual

identifications, as D*As = As, we have Lots = OSO0 where Do is the Laplacian acting on C°O(M). So, if Va is the eigenspace corresponding to the eigenvalue A, then

Va=VPo®...®V4 where

Aof=liif,fEV,,, and

Uk>...>u1> 40=0. It is immediate that µi are eigenvalues of As smaller than A. Moreover, if Va = Vim, for any f E Va there is an eigenfunction f* of d2/dt2 on S1 so that 7r* f * = f. In particular A = 0/b,2, k E Z. Suppose now that S is not

6.3. A SPECTRAL CHARACTERIZATION

65

simply connected. Then the eigenvalues of Do in the open interval (0, 4n) are of the form k2/bj2. Therefore, there is no eigenvalue with multiplicity 2n in this interval. Yet, by the assumption on the generator a = exp(- 20(b - 2a)) of CHQ, the eigenvalue a2/b2 + 2n - 1 of Ao has multiplicity 2n and belongs to the given interval. This contradicts the hypothesis on the equality of the real spectra on functions. Now we again apply Theorem 6.6 to conclude that M is biholomorphically isometric to Ma,&. The conclusion follows from

Lemma 6.2 Let CHQ be a complex Hopf manifold with generator as in the statement of Theorem 6.10. If Spec(CHQ, Do) = Spec(Ma,,b,, Do) and

Spec(CHQ, Do,o) = Spec(Ma,,b,, Co,o),

then Ma,,1, is biholomorphically isometric to CH,' or CHQ.

We begin by showing that b' = b and (aj)2 = a2 for any j. Indeed, as a first consequence of the equality of the real spectra, we have vol (Ma,,y) = vol (CHQ ). Since

vol (Ma,,,) = 27rvol(S2n-1(1)) vol (CHQ,') = 2irvol (S2i-1(1))

we get b = b'. On the other hand, the eigenvalues of Do on CHQ are of the form (p + q)(p + q + 2n - 2) +

p(a(p - q) - k)2

with p, q E Z+ and k E Z (computed by E. Bedford & T. Suwa, [201). Hence,

it is not difficult to see that, when b > (1/2) 2n + 1, the only eigenvalues of Ao on CHQ in (0, 4n) are a2/b2 + 2n - 1 (with multiplicity 2n) and k2/b2 (for some k E Z). Therefore, if we show that aj1 2/b12 + 2n - 1 and k2/b12 contained in (0,4n), then we may deduce are eigenvalues of Ao on (from the equality of the spectra of Do) that aj' 2 = a2 for any j. Here one uses the fact that harmonic polynomials of type (p, q) (i.e. polynomials of the form f(z) = F, IµI=P,IvI=9

CHAPTER 6. STRUCTURE THEOREMS

66

on C") give rise to eigenfunctions of Ao on C"- {0}. To be precise, if A E C, then

Do(IIzII''f) = (-A2 + 2(p + 4)(n - 1- A))IIzIl'f (cf. [201). Now, if A = -1+,f--la?/b, then IIzllAzi is Ga.,b-invariant. Hence it is an eigenfunction of Ao on Ma',b with eigenvalue (a,' )2/b2+2n-1. Similarly

for A = -1- a,/b. Also, for \ = -v/_-11 k/b, IIzlIA is Ga',b-invariant, etc. Finally, since b > (1/2)2n + 1, (a;)2 /b2 + 2n - 1 < 4n. , Q},1 < Q < n - 1, and a; = -Ian for j E If a j ' = Jai f o r j E { 1, , n} (a ¢ {0, -1/2}), then one can prove, using arguments similar {t + 1,

to the above, that Spec(Ma,,y, 00,0) j4 Spec(CHH, DO.o)

Thus Ma,,y is biholomorphically isometric to the complex Hopf manifold

with generator a or a. That is, the spectra of A0 and Do,o alone cannot distinguish between CHa and CH.'. The proof is complete. As we shall see in Chapter 8, for the case of generalized Hopf surfaces the equality of the spectra of Ao suffices to derive local isometry with the complex Hopf surface.

6.4

k-Vaisman manifolds

Let M be a Vaisman manifold and k E R - {0}. Then M is a k-Vaisman manifold if each leaf of the canonical foliation Fo of M has constant psectional curvature k. The idea is due to J.C. Marrero & J. Rocha, [1791, who use the term k-generalized Hopf manifold. It generalizes the notion of Vaisman manifold because, by Proposition 5.1, any g.H.o manifold is a c2Vaisman manifold, where c = 11w11/2. Although one drops the assumption that M has flat local Kahler metrics, by using a method of D. Janssens & L. Vanhecke, [1451, one may derive the explicit form of the curvature tensor field of M. Namely, one has

Proposition 6.1 Let (M, J, g) be a Vaisman manifold. Then (M, J, g) is a k- Vaisman manifold if and only if c2

R(X, Y, Z, W) = k k

4 c2 4

{g(Xo, Zo)g(Yo, Wo) - g(Yo, Zo)g(Xo, Wo)}

{v(X)v(Z)g(Yo, WO) - v(Y)v(Z)g(Xo, Wo)

+ v(Y)v(W)g(Xo, Zo) - v(X)v(W)g(Yo, Zo)

6.4. K-VAISMAN MANIFOLDS

67

+ g(Zo, JYo)g(JXo, Wo) - g(JXo, Zo)g(JYo, Wo)

+ 2g(Xo, JYo)g(JZo, Wo)}

(6.8)

for any X, Y, Z, W E X (M), where Xo, Yo, Zo, Wo denote respectively the T(Fo)-components of X, Y, Z, W with respect to T(M) = T(.Fo) (D T(.Fo)1, and c = IIwII/2. The proof of Proposition 6.1 follows, as .7 o has totally geodesic leaves, from Theorem 3.5 of [145]. As a corollary of Proposition 6.1, if M is a k-Vaisman

manifold and H(X) the holomorphic sectional curvature (i.e. the sectional curvature of the 2-plane spanned by {X, JX}), then

H(UU) = H(VV) = 0, H(X) = k

(6.9)

for any unit vector X E TT(M) which is orthogonal to U. and V. To give examples of k-Vaisman manifolds, let (N, cp, l;, 17, ry) be a c-Sasakian manifold

and (J, g) the almost Hermitian structure on M = N x R defined by

J(X,aat) =

(X+a_?i(X))

g((X,a'),(Y,b')) =ry(X,Y)+ab

(6.10)

for any X, Y E X (N) and any a, b E Coo (M). Then (M, J, g) is a Vaisman manifold with Lee form w = 2cdt and the Lee field B = 2c8/8t. Furthermore, if N has constant cp-sectional curvature k then (M, J, g) is a k-Vaisman manifold. Hence S2`1 (c, k) x R, R2t-1(c) x R and (R x Bn-1) (c, k) x R are k-Vaisman manifolds. Using Theorem 3.7 of [269] or Proposition 5.2 of this book one obtains

Corollary 6.1 (Cf. [179]) Let (M, J, g) be a 2n-dimensional complete kVaisman manifold. Let M be the universal covering space of M and (J, g) the induced Hermitian structure on M. Let c = iIwII/2. Then is holomorphically isometric to 1) if k > -3c2 then Stn-1(c, k) x R; 2) ff k = -3c2 then (M, J, g) is holomorphically isometric to R2n-1(c) x R; is holomorphically isometric to 3) i f k < -3c2 then ( M

(R x Bn-1)(c,k) x R.

Chapter 7

Harmonic and holomorphic forms In the present chapter we deal with harmonic forms and holomorphic forms and vector fields on compact Vaisman manifolds using the method in (275].

One of the main results is a partial answer to Vaisman's conjectures: A compact g.H. manifold has an odd first Betti number. We shall also study the relation between holomorphic and Killing vector fields and give a certain

answer to the question: How many l.c.K. metrics exist on a compact g.H. manifold?

7.1

Harmonic forms

Let (M2", J, g) be a compact g.H. manifold. We may assume JIwli = 1 so that the results of Chapter 6 apply with U = B, V = A, u = w, v = 0. For simplicity, let us also set, V= Dl ® D2 and call it the vertical foliation. Its properties were studied in Theorems 5.1 and 6.3. The direct sum decomposition T(M)=V ® Vl (which, on the local level, reflects the possibility of considering canonical cobases of the form {dza, &Z-a' w, B}, a = 1, ..., n - 1) produces a corresponding decomposition of the differential forms on M into sums of bihomogeneous forms of type (p, q), where p is the transversal degree and q the leaf degree. This, moreover, decomposes the exterior differentiation

operator as

d = d'+ d" + e, (7.1) where d' has type (1,0), d" has type (0,1) and 8 has type (2,-1). We say that a differential form 0 on M is V-foliate (or basic) if iAcb = iBO = 0 and GAO = LBO = 0. Equivalently, 0 is of type (p, 0), with 0 < p < 2n - 2, and

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CHAPTER 7. HARMONIC AND HOLOMORPHIC FORMS

d' o = 0. This means that 0 only depends on the leaves of V, i.e. in the local expression of 0 only za, T, dza, dz appear. In particular it is clear that the V-foliate forms Sl`(V) are preserved by d and thus determine a subcomplex of the De Rham complex. Let b, b', b", 8 be the adjoint operators of d, d, d", 8 respectively. Obviously 6' has type (-1, 0), b" has type (0, -1) and 8 has type (-2,1). As the Hodge * operator with respect to g acts homogeneously, the decomposition (7.1) yields a corresponding one for the adjoint operators (cf. I. Vaisman, [282]): 6

=(-1)P+a*-'d*=6'+b"+8.

(7.2)

Now let A be any r-form on M. One decomposes it uniquely as

A=a+wA/3, where, with respect to a canonical cobase, a and ,6 do not contain w. We want to investigate the properties of a and ,0 when A is harmonic. First of all we have

Lemma 7.1 (cf. [2751) A is harmonic if and only if a and /3 are harmonic.

Proof. Using the parallelism of w one can show that the Laplace-Beltrami operator preserves this decomposition in the following sense:

AA =&a+wA0/3,

(7.4)

This proves the "if' part. For the converse, taking the exterior product with w in both members of (7.4) yields w A Da = 0. Now we take the interior product iB with B and note that it commutes with A (cf. A. Lichnerowicz, [173], p.159). Thus Aa = 0. Similarly, z/3 = 0. Before going on we need the following result, similar (in statement and proof) to one known for K-contact manifolds (cf. [25], p. 69):

Lemma 7.2 (cf. [153]) If A is a harmonic r-form, 0 < r < n - 1, then iAA = 0.

This will be necessary in order to prove that a and (3 are V-foliate. It is enough to show that the Lie derivatives in the directions of A and B vanish.

Indeed, from the above lemma iAa = iA3 = 0. Since a and /3 are closed, this yields CAa = £A/3 = 0. Further, from (7.3) iBa = iB/3 = 0 which similarly implies LBa = LB/3 = 0. This shows, in particular, that in the case of Theorem 6.3, the forms a and /3 are projectable onto the Kahler base

M/V. The next step is

7.1. HARMONIC FORMS

71

Lemma 7.3 (cf. [275]) A harmonic V -foliate r -form is transversally harmonic, i.e. it is closed and coclosed with respect to the codifferential of the transversal part of the metric g. Proof. If da = 0, by type comparison in (7.1) we get d'a = 0. Similarly, from (7.2), ba = 0 implies b'a = 0 and as = 0. Let us denote with *' the

Hodge operator of the transversal part of g (i.e. of the first term in formula (5.6)). Then one can see that

*a=0AwA*'a. From the last assertion of Theorem 5.1 we know that dO = -igaEdza Adz This implies d'O = 0. Obviously d'w = 0. Hence

b'a = (-1)' *-1 d * a = (-1)' *'-1d' *' a, thus 8' is precisely the codifferential the statement refers to (note that this wasn't obvious in the definition of b'). The proof is complete. If we denote by A' = d'b' + b'd the basic Laplacian, what we just proved

is that 0'a = 0. The last step before stating the main structure theorem for harmonic forms is

Lemma 7.4 (cf. [2751) For a V-foliate r form a, 8a = 0 if and only if

idga=0. Note that because of the relation ) = dO - w A 0 (see the proof of Theorem 5.1), the last equality of the statement means that a is transversally effective, i.e. effective with respect to the first term of (5.6) (cf. [173] and [106]).

Proof. It is easily seen that as = 0 is equivalent to < aa, /I >= 0 for each /3 of bidegree (r - 2, 1) (where denotes the global scalar product of forms on the compact M). Due to the type of /8 we must have

,3 =0A/3j+-A02, 01 and /32 being orthogonal to w and 0 iAfi = iBf32 = 0

for i = 1, 2. Therefore /j1 = iA13, 00 = dO A,61 = dO A iA/3,

CHAPTER 7. HARMONIC AND HOLOMORPHIC FORMS

72

Finally < 8a, /3 >=< a, e/3 >=< a, e(dO)iA,Q >=< e(6)idea,Q >

where e is the exterior product operator. This means that 8a = 0 if and only if e(O)idoa = 0.

Altogether, the above lemmas prove

Theorem 7.1 (cf. [275]) Let (M2,, J, g) be a compact Vaisman manifold. Then any r -form A on M, 0 < r < n - 1 is harmonic if and only if \ = a + w A /3, where a and /3 are transversally harmonic and transversally effective V-foliate forms.

This result has important consequences concerning the topology of M. Let us denote by if (M, V) the vector space of V-foliate transversally harmonic r-forms (the basic cohomology groups), which are the cohomology groups of the complex (S2*(V),d) and by er(M,V) its real dimension. These spaces satisfy the Poincare duality because *' isomorphically identifies the spaces V). With these notations we have 9-lr(M,V) and

Theorem 7.2 (cf. [275]) On a compact Vaisman manifold the numbers er are finite and are related to the Betti numbers of M by the relations:

br=er+er_1-er_2-er_3 (O sr_2k < oo (0 < r < n - 1). k=0

(7.5)

7.1. HARMONIC FORMS

73

By Poincare duality we also have er < oo for n < r < 2n - 2. Clearly the other er vanish. Again from (7.5) we derive

er=Sr-Sr_2 (0

-2(n-r-1)

1(n-r)(n-r-1) . At this point we may state the complex form of Theorem 7.1:

Theorem 7.4 (cf. [262]) Let (M2n, J,g) be a compact Vaisman manifold. Then any (p, q)-forma on M, 0 - < A'a, Aa > 1

+ 2(r-n)



- < A'a, A'a > - 2 (n - r) < i7a, i7a > . Since, on the other hand, one has < 8i7a, a >

< i78a + e(7)a + e(7)ja, a > - < e(7)a, sa >= 0,

the following equation is obtained

+2(n-r) =0. Hence, for r n + 1. As for r = n, one uses the following formula for the Euler-Poincare characteristic (cf. [291], p. 151): 2n

2n

X(M) _ >(-1)rbr(M) = E

(-1)p+9hp.9(M)

p+q=o

r=O

Remark 7.3 Note, however, that hp,q(M) may differ from h4,p(M). E.g., from the above results we have h0,q(M) = h9-1'0(M)+hq.0(M) for q:5 n-1. Consequently h1"0 = 2(b1 - 1) while h°,1 = 2(bi + 1)

As to parallel forms on a compact g.H. manifold, they are very few. Indeed, one has

Theorem 7.7 Let M be a connected compact Vaisman manifold which is not Kahlerian. Then, if a is a parallel p-form on M then a = ku with

kER forp=1 anda=0 for2 3 then a,#, -y are constants (as in the case of a locally conformally flat g.H. manifold). This follows from

7.2

Holomorphic vector fields

In this section we shall denote by T1"0(M) the holomorphic tangent bundle

of M. Also, we shall identify Vl with the quotient bundle T(M)/V and let it : T(M) -+ Vl be the canonical projection. The image 1r(Z) will be denoted by [Z]. Let (V')"0 be the (1,0) subbundle of Vl 0 C. Then it restricts to a bundle homomorphism of holomorphic vector bundles, denoted by the same symbol, 7r : T1,0 (M) -+ (V1)1,o The obstruction for a to be an isomorphism is given in the following:

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CHAPTER 7. HARMONIC AND HOLOMORPHIC FORMS

Theorem 7.9 (cf. [275]) Let M be a (not necessarily compact) Vaisman manifold and Z a holomorphic section of V1. Then there exists a welldefined cohomology class [ic(Z)] E H1(M,OM), and Z = 7r(Z) for some holomorphic vector field Z on 11'1 if and only if [r.(Z)] = 0 (where OM is the sheaf of germs of holomorphic functions on M).

Proof. With respect to a canonical basis, {a/aza, Zo = B - /TA} is a local basis for T1"0(M). Next Z is a holomorphic vector field on M if and only if it has the local expression

Z=(aaza+fZ0 where (a and f are holomorphic functions. If we make a coordinate change of the form za = za(zb), in = in(xa,z"), (a and f must change as follows: (a =

azb

(b'j=f+1 (ail, ap) (a , a aza

/ az

where A is an analytic, nowhere zero, local function defined by Zo = A(za, z")

(a/azn). Clearly, if Z has the above local expression, then 7r(Z) is a holomorphic section of V1. Now let

Z=(a

[

z0

a].

If Z1, Z2 are differentiable vector fields of the form (7.7) and such that ir(Z1) = 7r(Z2) = Z (such Z; always exist), they must have same components

(a. Instead, fl - f2 must be a global holomorphic function V) on M (thus constant if M is compact). Hence a holomorphic Z with -7r(Z) = Z exists if it has the form (7.7) and there exists a global holomorphic function

on M such that f + 0 is a local holomorphic function: D f = -'50. On the other hand, from (7.8) we see that D j = a f . This means that the local 1-forms {-a f } define a global (0, 1)-form which we denote by a(Z). If 7r(Zl) = 7r(Z2), then rc(Z1) = rc(Z2)+av for some function v. Consequently, the a-class [k(Z)] is well defined and can be viewed as a class in Hl (M, OM). This ends the proof.

Corollary 7.3 (cf. [275]) If M is a compact Vaisman manifold, the only holomorphic vector fields in V are of the form cV, c E C. If, moreover, V1 has no nonvanishing holomorphic sections, these are the only holomorphic vector fields of M at all.

7.2. HOLOMORPHIC VECTOR FIELDS

81

On the other hand, on a compact M, for a given holomorphic section Z of Vl, K. Tsukada constructed a basic Dolbeault cohomology class [k(g)) E 9-l°"(M,V). We shall briefly sketch this construction. Let t; be the (0, 1) 1-form corresponding to Z. First, by direct computations, one shows that is '5-closed. Next we show is V-foliate. As O = 0, we have the decomposition l; = o+af, with OJo = 0 and f a complex-valued function on M. Recall that the complex Laplacian commutes with the inteis constant on i,,Deo = 0. Hence t o (_V) = rior product; thus M. We may write

vol(M)iA = M zA * 1 =< 6,17 > < 1; - acp >_
- < f, Ov >= 0.

Theorem 7.4 implies that to is V-foliate. Moreover, f is holomorphic on the

leaves of V: Indeed, Zo(f) = iVl; - iv6o = 0. But the universal covering spave of any leaf is, up to biholomorphisms, C. As f is bounded, it must be constant on each leaf. So f is V-foliate. In conclusion, 1; is a basic a-closed (0, 1)-form whose cohomology class in

p{°"(M,V) is denoted [k(2)). It can be seen that the relation -2[r.(Z)] = [k(Z)] holds in 9-l°"1(M,V). A similar interpretation for [,c(Z)] can be given:

Proposition 7.1 (cf. [262]) Let M be a compact Vaisman manifold and 2 a holomorphic section of Vl. Then there exists a holomorphic vector field Z on M such that Z = ir(Z) if and only if [k(Z)] = 0.

In particular, if bl(M) = 1, then h1,0(M) = 0 (see Remark 7.3) and 7r T",o(M) - (VI)l,o is an isomorphism. This is the case for compact homogeneous g.H. manifolds, for the Hopf manifolds with the Boothby metric or for the Inoue surface with the Tricerri metric. As in the Kahlerian case, imposing restrictions on the curvature of a compact g.H. manifold gives further information on holomorphic vector fields. We quote without proof two such results. To state them, let t be the complex Lie algebra of holomorphic vector fields and t its real subalgebra consisting of holomorphic vector fields whose associated real vector fields are Killing. Then

Theorem 7.10 (cf. [275]) If the transversal Ricci tensor of a compact g.H. manifold is negatively defined, then the only holomorphic Killing vector fields

are the real multiples of A and B.

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CHAPTER 7. HARMONIC AND HOLOMORPHIC FORMS

Theorem 7.11 (cf. [2621) On a compact g.H. manifold with constant scalar curvature we have

?3=R+VTl, lnv/---lR={cZoICEC}. We end this chapter by formulating an answer to the problem: How many l.c.K. metrics exist on a compact 1.c.K., not g.c.K., manifold (M, J,g)? The place of this result in this chapter is justified merely by the method of proof. Let M be a compact complex manifold which admits at least one 1.c.K., not g.c.K., metric. Let M be the set of all l.c.K. metrics on M. Each g E M determines, via its Lee form, a de Rham cohomology class l (g) E

l(g) E H' (M, R). We want to HI (M, R). Thus we have a map l : M determine the image of 1. To this end, let H be the subspace of H$'0(M) of d-closed holomorphic one-forms. Let j : H!'0(M) -i H1 (M, R) be the real injective linear map defined as j(a) = the de Rham cohomology class the real component of a belongs to. Let H = Im(j). From Theorem 2.1 in [271] we know that 1(M) n H = 0. We now specialize the discussion to a compact g.H. manifold (ll'i, J, go) with Lee form wo. By Theorem 7.5 one has 7H = He'0(M). Thus, dimR, H = 2 dims Ha ° (M) = 2h1'0.

Further, by Remark 7.3 one has bl = 2h1'0+1 hence dim H = dim H' (M, R)1. As [wo] ¢ H, an arbitrary l; E Hl (M, R) can be written as e = t[wo] + r/ with t E R, 11 E H. Now we are prepared to state

Theorem 7.12 (cf. [262]) Let (M, J, go) be a compact Vaisman manifold with Lee form wo. Then

1(M)={t[wo] +g[t>O,gEf}. Proof. We proceed by double inclusion. Let us first show "D". We assume the length of wo to be 1, so our proven results apply to Ao, Bo, etc. For a holomorphic one-form \, let a and ,0 be its real and imaginary part: 2a = (A-a). We know they are closed and V-foliated by Theorem 2(3 = 7.5. We now define a l.c.K. metric with parallel Lee form w = two + a, t > 0 arbitrary, by setting

g = t(go-w)®wo-oo®8o)+(two+a)®(two+a) +(too +,3) ® (too +#).

7.2. HOLOMORPHIC VECTOR FIELDS

83

That g is l.c.K. with Lee form w is a simple checking. To see that w is parallel one shows that the Lee field B = (1/t)Bo is Killing (here we use the foliated character of a, (3) and applies iii) of Proposition 4.2. For the converse inclusion we prove that for arbitary t > 0 and 17 E f,

-t[wo] + r ¢ 1(M). It suffices to see that no 1.c.K. g can exist with Lee form w = -wo. By absurdity, let g be such. Then the Lie derivative of the fundamental 2-form SZ of g in the direction B is easily seen to be £BO = -dOo + SZ - wo A Oo.

We want to show that the inner product (LBSZ, 1) is everywhere positive. This leads to the relation fLB1l'

f

=

fM

= n! fM(LBO,11) * 1 > 0 which contradicts the obvious relation

JM LBO.=

fdiBr=O.

Indeed, a straightforward computation shows that 2n

(LBSl, SZ)

2

deo(ei, Jet) + (n - 1)

where {e;, Je; } is a g-orthonormal basis. We can compute the first term in the right hand side using the relation -dOo = no - wo A 00, true on any g.H. manifold, and see that it is positive. Now the proof is complete.

Chapter 8

Hermitian surfaces In this chapter we decribe the l.c.K. surfaces among the Hermitian surfaces. Of course, we are motivated by having at hand examples such as Hopf and Inoue surfaces, which were previously seen to admit natural l.c.K. structures. On the other hand, in general, on each Hermitian surface the identity dil =w A Sl

holds, where the Lee form w is given by

w=(6S0)0J. Yet, in general, w is not closed (while, if it satisfies dit = w A S2, it is always closed in complex dimension n > 2). Let M be a Hermitian surface with Lee form w. In a local unitary frame {El, E2, E3, E4} (where E3 = JEl and E4 = JE2) the Lee form satisfies 4

E(VE;W)JE; = 0. =1

F. Tricerri & I. Vaisman (cf. [2601) classified the Hermitian surfaces according to certain properties of Vw. We now present this classification. Let (V, J, g) be a Hermitian vector space of real dimension 4. Consider

rr(V)=IT EV'®V`:FT(EI,JE;)=0} where {E;} is a fixed unitary basis of V. When V is the tangent space at some point of M, the space r(V) contains all tangent (0, 2)-tensors which behave like Vw. There is a natural action of U(2) on -r(V) given by (U - T) (vi, V2) = T(Uvi, Uv2).

CHAPTER 8. HERMITIAN SURFACES

86

We may decompose r(V) into a direct sum of irreducible components under this action. One such decomposition is

r(V) = ®a=ira(V)

(8.2)

where

rl(V)={TEr(V):T=Ag,\ER} r2(V)=IT Er(V):Vv,wEV, T(v, w) = T(w, v) = T(Jv, Jw), ET(EE,E,) =0} r3(V)

IT E r(V) : `dv,w E V, T(v,w) =T(w,v),

T(Jv,Jw) = -T(v,w)} ,r4 (V) _ IT E r(V) : Vv,w E V, T(v,w) _ -T(w,v), T(Jv, Jw) = T(v, w)}

r,(V)={TEr(V):Vv,wEV,T(v,w)=-T(w,v), T(Jv, Jw) = T(v,w)). Indeed, it is easy to check that the spaces ra(V) are U(2)-invariant and their sum is direct. The irreducibility follows from the Weyl invariant theory, taking into account that g(T, T) = IIT112 is the only quadratic form independent on each ra(V). We consider five classes of Hermitian surfaces, labelled from (1) to (5), with respect to the decomposition (8.2). That is, M E (i) if and only if for

anyxEMonehasT=(Vw).Er,(T,(M)). Remark 8.1 For Hermitian surfaces with nowhere vanishing Lee form, it has been shown (cf. P. Piccinni, [229]) that the decomposition (8.2) may be refined as follows. On such a surface the structure group U(2) may be reduced to H U(1) acting by transformations of the form

F1 =aE1+bE2i F2=-bE1+aE2 where {E1, E2} and {F1, F2} are (local) unitary frames in D3 (with E2 = JE1

and F2 = JF1) and B and JB are left invariant by H. Under the action of H, one may decompose r(Tx(M)) into ten irreducible components (see P. Piccinni, [229]).

8.

HERMITIAN SURFACES

87

We now look for analytic decriptions of the classes (i), 1 < i < 5. We

setA=-JB,0=woJ. Then

0 =6BSZ, w=-LAS2. The covariant derivative of the Kahler form is given by

2(Vx1)(Y, Z) = 6(Z)9(X,Y) - 0(Y)9(X, Z) + w(Z)SZ(X, Y) - w(Y)1(X, Z)

(8.3)

Set X = B in (8.3). We obtain VBSI = 0.

(8.4)

Next, one may use the relation between the Lie derivative, the interior product, and the exterior differentiation operator to derive

GBSl=do+k2Sl-wno, k=IIw112

(8.5)

GAS2 = -dw.

(8.6)

Also, the following identities are straightforward:

(CBSl)(X,Y) = (Vxw)JY - (Vyw)JX

(8.7)

(GB9)(X,Y) = (Vxw)Y + (VYw)X

(8.8)

(GAg)(X,Y) = (Vxw)JY + (VYw)JX.

(8.9)

By the identities (8.4)-(8.9) one obtains the following description of the classes (i):

M E (1) s Vw=Ag, 4)=bwEC°O(M)

ME (2).dw=0, bw=0, GAg=O ME(3)dw=0, LBS2=0 ME(4) LBg=0,GBQ=0 ME(5)'# GBg=O, GAg=O. Next, we are concerned with finding sufficient conditions under which a surface in class (i) is 1.c.K. or a g.H. manifold.

Theorem 8.1 Let (M,J,g) be a connected Hermitian surface in class (1). If M is compact or if it is not globally conformal Kdhler then M is a Vaisman manifold.

CHAPTER 8. HERMITIAN SURFACES

88

Proof. Following F. Tricerri & I. Vaisman, [260), we start by computing dA. We have

dw = 0, LB9 =2Ag, LBSl=2A1, d(k2-2A)AC = 0. As rank(Q) = 4, the last equation yields k2

- 2A = const.

on M. Then d(2A) = dk2 = d!Iw1I2 = dtBw = GBW = 2Aw.

Two possibilities arise. First, if A is nowhere zero on M, then w = dlog JAI. If this is the case then g' = (1/IAI)g is a Kiihler metric (thus contradicting

the assumption that M is not g.c.K.). If M is compact, then by Green's lemma we obtain

I

MA*1=-1 4

(Ew)*1=0.

fm As A has constant sign on M we obtain A = 0 hence the Lee form is parallel.

Secondly, let x E M with A(x) = 0. We set w = -dlogr with r defined in some neighborhood U of x. Then, as above, we have kr = const. on U. Hence, as A(x) = 0, we get A = 0 on U. As M is connected we conclude that A = 0 on M and the proof is complete. Thus, a connected, noncompact Hermitian surface M in the class (1) is g.c.K. or a g.H. manifold. A natural question is: Are there any non g.H. Hermitian surfaces in class (1)? When A is nowhere vanishing, let g' be the conformal Kahler metric as above, and D its Levi-Civita connection. One has

DdA= (A2IAI - AIldAII2)gl+ 2dA®dA.

Suitable contraction furnishes p9,,\ = -4A2IAI.

(8.10)

It is an open problem whether (8.10) admits nontrivial solutions. A solution of (8.10) would provide a surface in class (1) with nonparallel Lee form. As to classes (2), (4) and (5), we have the following:

Theorem 8.2 Let (M, J, g) be a compact Hermitian surface in class (i) urith i E {2,4,5}. Then M is a Vaisman manifold.

8.

HERMITIAN SURFACES

89

Proof We follow F. Tricerri & I. Vaisman, [260] and T. Kashiwada, [150]. For M E (2) and M E (5) the proofs are technical and rely essentially on the computation of the Laplacian of the Lee form. If M E (5) then (8.9) leads to

(Vjxw)JY = -(Vxw)Y -(Vxvyw)Z - (V(vXJ)yw)JZ+ (Vyw)((VxJ)JZ). On the other hand, by definition

Ow=-EVE, VE,w+S(B, ) i

and this yields Ow = -2VBw = 8dw.

Yet (VBw)B = g(w, dk2) hence by integrating over M

fg(w,w)* 1

-JMg(w,dk2)*1=JMg(6w,k2)*1=0.

Consequently dw = 0 and M is 1.c.K. As B is Killing M is actually a g.H. manifold.

If M E (2) the vector fields A and B are holomorphic. One can now show that for an analytic vector field H and for its dual (i.e. H = a) the following formula holds on a Hermitian surface:

(OC)X = 2S(H, X) - (8£)w(X ) -(1.Hw)X - 2g([B, H], X). The proof (cf. F. Tricerri & I. Vaisman, [2601) is too technical to be reproduced here. In particular for H = B (and = w) we obtain

2S(B, B) = 2(VBw)B = g(w, dk2). As above, after integration over M, one gets JM

S(B,B)*1=0.

Finally, by Bochner's formula, Vw = 0 and the proof is complete. As to class (4), by a result in T. Kashiwada, [150], a skew symmetric 2-tensor field T satisfies

T(JX, JY) = T(X, Y)

,

A(Sl)T = 0

CHAPTER 8. HERMITIAN SURFACES

90

if and only if *T = -T. Now let T = Vw and take into acount that dw = 2Vw within class (4). This yields *dw = -dw or, equivalently, b * w = -dw which gives (after integration over M) dw = 0. The proof is complete.

As to the noncompact surfaces lying in classes (1), (2) and (5), a few characterizations may be formulated in terms of certain distributions naturally arising on these surfaces under the assumption that w # 0 everywhere. To illustrate our ideas, we restrict ourselves to surfaces in class (1), yet we remark that similar results are already available for surfaces in the remaining classes, cf. P. Piccinni, [229].

Theorem 8.3 Let (M, J, g) be a Hermitian surface lying in class (1) and having a nowhere vanishing Lee form. Then the distributions D2 ® D3 and D1(D D2 are integrable. Let F0 be the foliation determined by D2 ®D3. Then

.ro is a totally umbilical Riemannian foliation, D' is conformal and totally geodesic, D19 D2 is totally geodesic, almost complex, and its leaves are real surfaces of Gaussian curvature K = -A. Moreover, the folowing assertions are equivalent: 1) Fo is totally geodesic; 2) V1 is Riemannian; 3) V 2 is totally geodesic; 4) Dl (D V2 is Riemannian;

5) K = 0; 6)M is a Vaisman manifold. Proof. First, we verify the properties of D2 ®D3 and D' and the equivalences 2 6. We showed (during the proof of Theorem 8.1) that Vw = Ag 1 implies dw = 0, hence D2 ® D3 is integrable. For any X E T(Fo) we have

g(VBB, X) = (VBW)X = Ag(B,X) = 0 hence D1 is geodesic and F0 is Riemannian. Let us now compute the second

fundamental form of F0. Let 7r : T(M) - Dl be the natural projection. Then

h(X,Y) = rVxY = kw(OxY)Bo = -k9(X,Y)Bo for any X, Y E Fo, where kBo = B (and kAo = A). On the other hand, using the (local) frame

{E=E1, JE=E2, Ao=E3, Bo=E4},

8.

HERMITIAN SURFACES

91

we may compute the mean curvature vector of a leaf of F0 as follows: lA =

3

{9(V EE, Bo) + 9(V JEJE, Bo) + 9(V AOAo, Bo)}Bo

_ - 13k{(DEw)E + (VJEW)JE+ (VAow)Ao}Bo A

3k

A

{9(E, E) + g(JE, JE) + g(Ao, Ao)}Bo = kBo-

Thus h = g ® µ, that is each leaf of Fo is totally umbilical. Consequently A=0 h=0 Vw = 0 and this proves the equivalence of statements 1, 2 and 6. Next, we show that there exist local functions au : U -+ R, with U C M open, so that Dl is Riemannian with respect to the metrics g' = exp(2ou)g. This will prove that D' is a conformal foliation. Indeed, we have d(2A) = 2Aw

so that the 1-form W = (A/k2)w is closed. Let Wu be a (local) solution of &pu = -cp. Let V° be given by

VXZ =

7r[X, Z] { irVXZ

if X E T(.)o)1 if X E T(.F'o).

Then

(V g')(X,Y) = B(exp(2ou))9(X,Y) + exp(2ou)(V g)(X,Y) = -2exp(2ou)A9(X,Y) - 2exp(ou)9(B,h(X,Y)) = 0. To conclude that Dl is Riemannian we need to compute the second fundamental form of its orthogonal distribution D2 ®D3. It clearly vanishes. Thus 9(VAA, B) = W(VAA) = -(DAW)A = -Ag(A, A). Consequently B 1 VAA M' a C°° map. Let V, V' be two linear connections on M, M' respectively. Let f'TM' M

be the pullback bundle of T(M') via f and f'v' the connection in f'TM' induced by v'. This is most easily described in local coordinates, as follows. Let (U, x') and (U', yCZ) be local coordinates on M, M' respectively, so that f (U) C U'. Let

y,, : f-1(U') -, f*TM'

Y.W =

a I ay,

, x E f-1(U') f (t)

be the natural lift of a/ay°. Then (f`v')818.; Y.y = If (r"0' of) Y. -Y

CHAPTER 9. HOLOMORPHIC MAPS

108

Here

fQ=yQ°f, f°= axi.

Also r'ay are the coefficients of V' with respect to (U', ya). The differential df of f may be thought of as a cross-section in the vector bundle T*(M) f *TM'. As such, it has the local expression df = f; dx' ®YQ.

Let of be the connection in T* (M) ® f`TM' defined by Vf (w ®s) = (vxw) ®s + w 0 (f `O')xs

for any 1-form w on M, any cross-section s in f`TM', and any X E X(M).

Let off; be defined by va/axj df = (V f f;) dx' Then Valazjdx' = -rjikdxk yields

v;f;=fi-r%f +(r'a7of)f°f;, a2f0/ax'ax2. We shall need the local expression of the second order covariant derivative of df. Define vkvf f ° by setting where

va/axk Vf df) = (vkvf f") dx& ®dx` ®YQ. Then

vkvff°=exk(vff;)-rk;vffa

-rkiV. fa+r'6-1 kvf fi. As a consequence of (9.3) one may derive the following commutation formula

vhvk fQ - vkvhf; _ -flQRthki+ fk fi fhR'Pdr,

(9.4)

where R, R' are the curvature tensor fields of V, V' respectively. Now let g be a Riemannian metric on M. The pseudotension field of f with respect to (g, V, V') is the cross-section -r (f ; g, V, V') in f *TM' defined by

r(f;g,v,v') =TQ(f;9,v,v')Y.

9.2. PSEUDOHARMONIC MAPS Ta (f ; g,

109

O, V') = g`jV f °.

Then f is pseudoharmonic with respect to the data (g, V, V') if r(f ; g, V, V') = 0. If g' is a Riemannian metric on M' and V, V' are the Levi-Civita connections of (M, g), (M', g') respectively, then T (f ; g, V, 0') is the usual tension field T(f) of f, and f is harmonic if r(f) = 0, see e.g. [91]. We may establish the following

Proposition 9.3 (Cf. [277]) Let M, M' be two complex manifolds and V, V' two torsion free almost complex linear connections on M, M' respectively. Let g be a Hermitian metric on M. Then any holomorphic map f : M - M' is pseudoharmonic with respect to (g, V, V').

Proof. Let (U, z') and (U', wa) be systems of local complex coordinates on M, M' respectively. Set

fa=waof , fa=fa. As f is holomorphic we have, by (9.1), '9fa

8z' Next

=

0.

r=r =rk=o ii ii

r' ,=ria =r'a7=0

(9.6)

as a consequence of VJ = 0 and Tp = 0 (respectively of V'J' = 0 and TV, = 0). As df may be extended by C-linearity to the complexification of T*(M) ® f `TM', it has the local expression, by (9.5) df = f," dz` c9 Wa + f dz` ®Wa, where

afa

f; = azi

and Wa is the natural lift of 9/,9wa to (f `TM') ® C. Then, by (9.5)-(9.6), we have

7

df

}dzk ® Wry. - { 8zk - rjk.fi + rpaff f;

(9.7)

CHAPTER 9. HOLOMORPHIC MAPS

110

Let B be a C°O(M)-bilinear form on M. Let {Ei} be a local orthonormal frame on (M, g). We set trace(B) = Ei B(Ei, Ei). With these notations

T(f;g,V,V') = trace(Vfdf). As the trace of an endomorphism of a real linear space V and the trace of its C-linear extension to V OR C actually coincide, we may compute T (f ; g, V, V') as follows

T(f;g,V,O')=A=+A Ai hjk = (v8/8zi df) azk = 0,

by (9.7), where hi; = g(a/azi,a/azj).

Remark 9.2 Proposition 9.3 includes the well known result that any holomorphic map between Kahler manifolds is harmonic (cf. [921) and that any holomorphic map between arbitrary complex manifolds satisfies an elliptic system of PDEs.

Corollary 9.2 Let (M,g) be a Hermitian surface and (M',g') a Kahler manifold (or another Hermitian surface). Let f : M --+ M' be a holomorphic

map. Then r(f;g,D,V') = 0 (orr(f;g,D,D') =0). Here D is the Weyl connection of M (given by (1.5)) and V' is the LeviCivita connection of (M', g') (respectively, if M' is a Hermitian surface then D' is its Weyl connection).

Lemma 9.3 Let (M, g) be a Hermitian surface and (M', g') a Kahler manifold. Let f : M -+ M' be a C°O map. Then

Ta(f;g,D,V') = ra(f;g,O,V') - fjBj.

(9.8)

Here V is the Levi-Civita connection of (M, g) while w = (%1) o J is the

Lee form of M (and B? = gijwi). To prove Lemma 9.5, let Df be the connection induced in T' (M) ®f'TM' by the pair (D, V') and let D? f ° be the corresponding covariant derivative (given by (9.2) where riki are replaced the coefficients of the Weyl connection D of M). Then (9.8) may be by obtained from r'(f;g,D,V')=gijD fa

and, by (1.5) from

r

S3

= I'S3

-

{wib + w3-b1

- gijBk}.

Using Lemma 9.3 one easily proves the following

9.3. A SCHWARZ LEMMA

111

Theorem 9.2 Let M be a Hermitian surface and M' a Kahler manifold. Let f : M -+ M' be a holomorphic map. Then f is harmonic if and only if Bx E Ker(dx f) for any x E M. The same result holds if M is a 1.c.K. manifold of arbitrary dimension. It generalizes the well known result by A. Lichnerowicz, [171], that a holomor-

phic map between Kahler manifolds is harmonic. This actually holds for almost Kahler manifolds, as well. A further generalization is S. Salamon's Theorem 4.2 in [241].

Let (M, g) and (M', g') be two Riemannian manifolds and f : M -' M' a COO map. The Riemannian metrics g, g' induce a Riemannian bundle metric

(,) in T* (M) ® f*TM' given by: (dx' (9 Y., d-%J ®Yp) = 9`j9' p

Set e(f) _ (cf, df) = II df 112. Locally

e(f) = 9'39apf*f' We shall need the following inequality

e(f)gII2,

(9.9)

for any C E T(M). This shows that e(f) characterizes the behaviour of lengths. To prove (9.9) we consider the cross-sections s = df and

r=9ijVfklkdx'®Ya in T*(M) ® f`TM', where

= C'a/ax'. Then (s,r) = II(df)e112

Ilsll = Ildf 11, 11r11= IICII II(df)EII

I(s,r)I P + 2 IIwII2 - (GB9) (C 0

Let M' be a complex manifold which has a Kdhler metric whose sectional curvature is bounded above by a constant K < 0. Then, for any holomorphic map f : M -* M' one has

2P K If P > K/2 then f is distance decreasing. If P > 0 and M is connected, then f is a constant map. Ildf II2
0 and any b < sup cp < +oo there is a sequence (xn)n>1 of points of M so that

h P(xn) = supcp lim lim n--

IW(xn) - 611+a

0

OW(xn) 511+2a - 0.

In particular, if sup W < +oo then there is a sequence (xn)n>1 of points of

M so that

sup ,

9.3. A SCHWARZ LEMMA

113

urn IIdWHI(xn) = 0 n-oo nlim00(i

)(xn) > 0

Let us show how (9.10) and Theorem 9.4 may be used to prove Theorem 9.3. By the asumptions i)-ii) in Theorem 9.3, (M, g) has Ricci curvature bounded below. Hence we may apply Theorem 9.4 (the Omori-Yau maximum principle) to the function cp = e(f) with a = 1/2 and b < sup e(f) in order to choose a sequence (xn)n>1 of points of M so that

urn e(f)(xn) = supe(f) 1im

Ilde(f)II(xn)

=0

n-°° I e(f) (xn) - 51312

l---

re(f) (xn) bI2 -

o.

Consider the function i& : M R given by

2Pe(f) - Ke(f)2 Ie(f) - 611

Then either supe(f) = +oo, and then zli(x,,) - -K for n

oo, or supe(f) < +oo, and then V1(xn) -+ (2PE - KE2)/I E - 6I2 for n -4 oo, where E = sup e(f ). So L = limn-,* 7P(xn) exists. However, we may show that L < 0 hence the first case does not occur. Indeed, by (9.10) we have

Ae(f)(xn) < 0 n-oo I e(f)(xn) - b12 -

L < lim (iwde(f)) (xn) - lim n-'°° I e(f)(xn) - 612

To derive the last inequality we also used i,,d e(f) = g(w, d e(f )) and then I(z,,de(f))(xn)l < IIwli(xn)Ilde(f)ll(xn) -+0 l e(f)(xn) - 612 Ie(f)(xn) - 612

for n -+ oo. Therefore E < +oo (otherwise L = -K > 0, a contradiction) and L < 0 yields

2P-KE 0 and then E = 0 (hence df = 0) or P < 0 and then II df Il2 -3c2

Let z :

S2n-1 -,

CPn-1 be the Hopf fibration. Let us denote by

the 1-Sasakian structure of S2i-1(1,1), i.e. the standard Sasakian structure of S2i-1. Then f is a Riemannian submersion of (S2r-1(1,1),')') onto CPn-1(4) (the (n - 1)-dimensional complex projective space of constant holomorphic sectional curvature 4), cf. also Appendix B. Actually J'o(dF) = (df)ocp', where J' is the complex structure on CPn-1(4). On any c-Sasakian manifold the integral curves of the contact vector are geodesics (cf. [145])

hence T has totally geodesic fibres. Now, let c, k E R, c # 0, k > -3c2. Then f may be also seen as a Riemannian submersion with totally geodesic fibres of the c-Sasakian manifold S2i-1 (c, k) onto CPn'1(k + 3c2) (the complex projective space of constant holomorphic sectional curvature k + 3c2). Moreover, if it is the projection on the first factor of the product manifold

S2n-1(c,k) x R then r(c, k, n) = T o Tr : Stn-1(c, k) x R -' CPt-1(k + 3c2)

is an almost Hermitian submersion with totally geodesic fibres of the kVaisman manifold S2n-1(c, k) x R onto the Kahler manifold CPi-1(k+3c2).

Remark 10.1 (Cf. [179)) If k > -3c2 then the Hermitian structure of S2n'1(c,k) x R induces a Hermitian structure on S"-'(c, k) x S' which

CHAPTER 10. L.C.K. SUBMERSIONS

126

makes S2i-1 (c, k) x Sl into a (compact) k-Vaisman manifold. If this is the case, the 1.c.K. submersion T (c, k, n) induces a l.c.K. submersion:

T(c, k, n) : S2n-1(c, k) x S' - CPn-1(k + 3c2)

with totally geodesic fibres. Note also that

S2n-1(c, k)

x S' is a compact

strongly regular Vaisman manifold. By Theorem 6.3 any compact connected strongly regular Vaisman manifold is a principal TC-bundle over a Ki filer manifold; in the case at hand, the corresponding Kahler manifold is

rs2n-1(c,k) x S') /AD' ®Dl)

10.2.2

CPn-1(k+3c2).

An almost Hermitian submersion with total space Rzn-1(c) x R

Let c E R, c # 0, and let (J, g) be the g.H. manifold structure of R2n-1(c) X R. Let (xi, yi, t) be the natural coordinates on R2r-1 (1 < i < n -1). Then

Xi=axi,Y= is a global frame tangent to

y

ii+2cx '

ata

09

,Z

at

R2n-1. The corresponding coframe is given by n-1

ai = dx` , ,3i = dy' , A = dt - 2c > xidyi. Next,

JXi=Y, JYi=-Xi,JZ=-as,J2s =Z g

n-1

jai (9 i=1

where s is the natural coordinate on R. Then {Xi, Yi, Z, 0/0s) is an orthonormal frame on (R2n, g) and the Levi-Civita connection V on (R2n, g) is expressed by

VX'Y = -DYXi = cZ VX;Z = VzXi = -cY VY;Z = VzYi = cXi (the remaining covariant derivatives vanish). Now consider the submersion 7r(c, n, m) : R2n-1(c) x R -> C'n(0) , m < n - 1

10.2. L.C.K. SUBMERSIONS

127

.r(c,n,m)( ...,x n-1 ,y ,...,yn-1 ,t,s) = (xI ,...,x m,y >1,, ,ym ). 1

1

Then 7r(c, n, m) is an almost Hermitian submersion of the Vaisman manifold (R2i-1 x R, J, g) onto the Kahler manifold Cm(0), having totally geodesic fibres.

Remark 10.2 (Cf. [179]) If c = 1/2 and H(n - 1,1) is the generalized Heisenberg group (cf. our Section 3.3) then the -3c2-Vaisman manifold R2n-1 (c) x R is holomorphically isometric to the g.c.K. manifold H(n 1, 1) x R, cf. 168. Let r(n - 1, 1) be the subgroup of H(n - 1, 1) consisting of matrices with integer entries. Then the Hermitian structure of H(n 1, 1) x R is invariant under the action of r(n - 1,1) x Z hence M(n, 1) = (H(n - 1,1) x R) /(r(n - 1,1) x Z) (i.e. N(n-1, 1) x S1 with the notations in Section 3.3) is a compact -3c2-Vaisman manifold and the submersion

7r(c, n, m) induces a l.c.K. submersion

ir(c, n, m) : M(n, l) - T2m of M(n, 1) onto the real 2m-dimensional torus T2' with the usual Kahler structure. The manifold M(n, 1) is also a compact strongly regular Vaisman manifold whose corresponding (in the sense of Theorem 6.3) Kahler manifold is

M(n,1) /( D1 ®D2)

T2n-2

i.e. the real (2n - 2)-dimensional torus.

10.2.3

An almost Hermitian submersion with total space (R x Bn-1)(c, k) x R, k < -3c2

Let (M', J', g') be a Kahler manifold with Kahler 2-form Q and let c E

R, c # 0. Assume that c Q is exact. Let us endow N = M x R with the c-Sasakian structure (5.3). Then M = N x R is a Vaisman manifold with the Hermitian structure (J, g) given by (6.10). On the other hand, the natural projection a : M -4 M' is an almost Hermitian submersion. The vertical distribution of 7r is spanned by (B, JB}, where B is the Lee field of (M, J, g). Since B is parallel, it has totally geodesic fibres. Let k E R, k < -3c2. Then, one may apply the previous considerations to the k-Vaisman manifold M = (R x Bn-1)(c, k) x R (if this is the case then M' = Bn-1(k+3c2) and N = (R x Bn-1)(c, k)) to conclude that the natural projection ry(c,k,n) : (R x Bt-1)(c,k)

x R-* B"-'(k +c2)

CHAPTER 10. L.C.K. SUBMERSIONS

128

is an almost Hermitian submersion with totally geodesic fibres of the kVaisman manifold (R x B'-')(c, k) x R onto the Kahler manifold Br-1(k+ 3c2).

10.3

Compact total space

In this section we look at l.c.K. submersions whose total space is a compact manifold and, under suitable assumptions, we relate the Betti numbers of the total space to those of the base manifold. A l.c.K. submersion possesses the following elementary properties (cf. (179], p. 279-280):

Proposition 10.1 Let 7r : M

M' be a Lc.K. submersion and w,w' the Lee forms of M, M' respectively. Then

w(X)=w'(X')o7r for any basic vector field X E X(M) which is zr-related to X' E X(M'). In particular, if B, B' are the Lee fields of M, M' then h(B) is a basic vector field ir-related to B'. The proof follows from (1.4) and (10.2).

Proposition 10.2 Let it : M -+ M' be a l.c.K. submersion. The following statements are equivalent: 1) the Lee vector field of M is vertical; 2) the fibres of 7r are minimal submanifolds of M; 3) the base M' is a Kahler manifold.

The proof of Proposition 10.2 follows from the fact that given a complex submanifold of M, the normal component nor(B) of the Lee field is (up to a constant) the mean curvature vector of the submanifold (cf. Theorem 5.1 of [275 or Theorem 12.1 of this book). In particular, Proposition 10.2 yields Theorem 10.1.

Proposition 10.3 Let it : (M, J, g) -, (M', J', g') be a l.c.K. submersion. Let A be the integrability tensor of it, 11 the Kahler 2 -form and B the Lee field of (M, J, g). Then AXY =

-1 fl(X,Y)JvB

for all horizontal vector fields X, Y on M. In particular, B is horizontal if and only if the horizontal distribution It of it is integrable.

10.3. COMPACT TOTAL SPACE

129

Corollary 10.1 The horizontal distributions of the l.c.K. submersions -r (c, k, n) : Stn-I (c, k) x R -+ CPn-I (k + 3c2) , k > -3c2 7r (c, n, m) : R2n-I (c) x R --+ Cm(0) , m < n - 1

y(c,k,n) : (R x Bn-I)(c,k) x R - Bn-I(k+3c2) , k < -3c2 are not integrable.

: M -+ M' be a C°° map between two Riemannian manifolds (M, g) and (M', g'). Let 6 (respectively 6') be the codifferential on (M, g) (respectively on (M', g')). Then, by a result of B. Watson, [284], ir* o b' = b o 7r' on 1-forms if and only if 7r is a Riemannian submersion with minimal fibres. On the other hand, as the Lee form of a Vaisman manifold is parallel, it is harmonic. Using these facts, Proposition 10.2, and Poincare duality, one may establish Let -7r

Proposition 10.4 (Cf. [179]) Let 1r : M --+ M' be a l.c.K. submersion so that the Lee vector of M is vertical and M is compact. Let dime M = 2n and dimR M' = 2m. Then the Betti numbers of M, M' are related by

b1(M) ? bi(M'), b2n-1(M) ? b2.-1(M'). If, in addition, M is a Vaisman manifold, then b1(M) > b1(M') + 1 b2n-1(M)

b2m-1(M') + 1.

Theorem 10.2 (Cf. [179]) Let it M -+ M' be a 1.c.K. submersion with totally geodesic fibres and M compact. Let dimR M = 2n and dimR M' _ 2m. Then b?(M) > bj(M') - bj_2(M') , 2 < j < m bj (M) > b2(m-n)+j (M') - b2(m-n+l)+j (M') , 2n - m < j < 2n - 2.

If in addition M is a Vaisman manifold then b2 (M) > b2 (M') + b, (M') - 1

bj(M) > bj(M') + bj-1(M') - bj_2(M') - bj-3(M') , 3
b2(m_n)+j (M') + b2(m-n)+j+l (M')

-b2(m-n+1)+j(M') - b2(m-n+I)+j+l(M') , 2n - m < j:5 2n - 3 b2n-2(M) >- b2m_2(M') + b2m-I(M') - 1.

CHAPTER 10. L.C.K. SUBMERSIONS

130

Total space a g.H. manifold

10.4

The 1.c.K. submersions from a Vaisman manifold possess the following elementary properties (cf. [179], p. 284-286):

Proposition 10.5 Let M be a Vaisman manifold and it : M M' a l.c.K. submersion. Then M' is a 1. c. K. manifold with parallel Lee form w'.

The proof follows from DJ = 0, the identity (10.2), and Propositions 10.1 and 10.3. Clearly Proposition 10.5 does not exclude the possibility that w' = 0 (i.e. that M' be a Kahler manifold).

Proposition 10.6 Let it : M --* M' be a l.c.K. submersion whose total space M is a Vaisman manifold with dim(M) > dim(M'). Then the horizontal distribution 1{ of it is not integrable.

The reason is that, due to the assumption on dimensions, the unit Lee field U of M fails to be horizontal. Let 7r : (M, J, g) -+ (M', J', g') be an almost Hermitian submersion. Let H, H' be the holomorphic sectional curvatures of M, M' and A the integrability tensor of it. Then, by a result of B. Watson, 1285], we have

H(X) = H'(X') - 3IIXII-4IIAx JX112 for any horizontal vector X E Rx where X' = (d ir)X (cf. Theorem 5.6 of [285]). Using this identity, Propositions 10.2 and 10.3, and Corollary 6.1, one may prove

Proposition 10.7 Let it : M M' be a l.c.K. submersion with minimal fibres and total space M a k- Vaisman manifold. Then M' is a Kahler manifold of constant holomorphic sectional curvature k + 3c2, where c = IIwII/2 and w is the Lee form of M. (Mj, Jj, gj) -. (Mj, Jj, gj) , j E {1, 2}, be two almost Hermitian submersions. We say that 7r1 and 9r2 are equivalent (notationally irl ir2) if there are almost complex isometries rr : (M1, J1, gl) , (M2, J2, g2) and z' : (M,', J1', gi) -' (M2, J2, g2) so that T' o 7rl = 7r2 o T. Then we may state Let 7rj

:

Theorem 10.3 (Cf. [179]) Let it : M -+ M' be an almost Hermitian submersion whose total space M is a simply connected complete k-Vaisman manifold with Lee form w. Assume that it has connected totally geodesic fibres. Let c = IIwIl/2, dim(M) = 2n and dim(M') = 2m. Then 1) if k > -3c2 then it - -r (c, k, n); 2) if k = -3c2 then it - 7r(c, n, m); 3) if k < -3c2 then 7r - ry(c, k, n).

10.4. TOTAL SPACE A G.H. MANIFOLD

131

If we make no assumption on the metric structure of M' but we assume M is a compact g.H. manifold we still can prove (cf. K. Tsukada, [263])

Theorem 10.4 Let 7r : M - M' be a holomorphic submersion of a compact generalized Hopf manifold onto a compact complex manifold. Assume

dime M > dime M'. Then M' bears a Kahler metric. We shall only sketch the proof. By assumption, each fibre 7r-'(p) is a complex submanifold of M of dimension k = dime M - dime M'. Then, by adapting a result of K. Abe, cf. [2], one can show that the fibres are foliated submanifolds; that is, for any p E 7r-'(p) the leaf through p of V is contained in 7r (p'). This implies (see Theorem 12.1) that the fibers are minimal in M. Let T (F) be the subbundle of T (M) consisting of the vectors tangent to the fibres and T(F)1 its orthogonal complement in T(M). Let ft be the restriction of Il to T(F) and consider the volume form XF on T(F) given by XF = (1/k!)S2k. If 77 : T(M) -+ T(F) is the orthogonal projection, then rl*XF is a 2k form on M which we denote by the same symbol XF. Now we proceed in constructing the announced Kahler metric on M'. For a vector v E Tp, (M') and p E 7r (p), let v* be the unique vector in Tp(F)1 satisfying d7r(v*) = v. Let gp(v*, w*)XF

9''i (v' w) = f-1W)

Clearly g' is a smooth Hermitian metric on M'. Its fundamental two-form is related to the one of g by QP',(v,w) n

I

np(v*,w*)XF (W')

To show that Sl' is closed, we first compute

Z*cp(X*,Y*)XF+

Z(H(X,Y)) =

stp(X*,Y*)GZXF

for vector fields X, Y, Z on M'. As the fibers are minimal, we have GZXF =

0 for each section Z of T(F)1 (cf. (6.17) in [257]), thus Z(Sl(X,Y)) = Jf-1(P')

Y'([X,Y],Z) =

Z*Ilp(X*,Y*)XF. Next

f

-1(p')

f?([X, Yl*, Z*)XF = f

I([X*,Y*], Z*)XF.

CHAPTER 10. L.C.K. SUBMERSIONS

132

Hence

dfl'(X,Y,Z) =

dfl(X*, Y*, Z*)XF w A Sl(X*, Y*, Z*)XF = 0

because B is tangent to the fibres. The proof is complete.

Chapter 11

L.c. hyperKahler manifolds In this chapter we show how some of the techniques previously introduced apply in the framework of Hermitian quaternionic geometry. To begin with, we recall the following terminology. Let M be a 4n-dimensional C°° manifold. A triple It, 12i 13 of global integrable complex structures on M satisfying the quaternionic identities, I,, 1a = Iy for (a,#, -y) = (1, 2, 3) and cyclic permutations, defines a hypercomplex structure on M. If a Riemannian metric g is added, assumed to be Hermitian with respect to 11,12,13, one gets a hyperhermitian manifold (M, g, 11, 12, I3).

More generally, by (M, g, H) we denote a quaternion Hermitian manifold. Here H is a rank 3 subbundle of End(TM), locally spanned by (not necessarily integrable) almost complex structures It, 12, 13, again satisfying the quaternionic identities and related on the intersections of trivializing open

sets by matrices of SO(3). H defines on M a structure of quaternionic manifold and the local almost complex structures 11,12,13 are said to be compatible with the quaternionic structure H. It is worth emphasizing that there exist quaternionic manifolds with the bundle H without any global section. The quaternionic projective space is an example. The additional datum of a metric g, Hermitian with respect to the local compatible almost complex structures, defines the quaternion Hermitian manifold (M, g, H). In Hermitian quaternionic geometry one has two corespondents for the Kahler condition. Namely, the hyperhermitian or quaternion Hermitian metric g is said to be hyperKahler or quaternion Kahler if its Levi-Civita connection V satisfies VI,, = 0(a = 1, 2,3) or VH C H respectively. We can now define (cf. [226]) the two classes of manifolds we shall be concerned with.

Definition 11.1 (i) A hyperhermitian manifold (M, g, It, 12, 13) is locally conformal hyperKdhler (l. c. h. K.) if, over open neighbourhoods {U;} covering

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M, glu; = efig' where g; is a hyperKahler metric defined on U;.

(ii) A quaternion Hermitian manifold (M,g,H), is locally conformal quaternion Kahler (l.c.q.K.) if, over open neighbourhoods {U;} covering M, glu; = ef'g' with g; quaternion Kahler on U.

Characterizations of 1.c.q.K. manifolds among the quaternionic Hermitian manifolds, in terms of differential ideals, are given by E. Bonan, [35], and by D.V. Alekseevskij & E. Bonan & S. Marchiafava, [4]. Let us note that 1.c.h.K. manifolds are also called hyperhermitian Weyl manifolds while l.c.q.K. manifolds are known also as quaternion Hermitian Weyl manifolds. For the first case the quaternionic Hopf manifold HH" = (H" - o)/r2 where 172 is the cyclic group generated by the transformation (hl,..., h") - (2h1, ..., 2h") (diffeomorfic to S' X S4"+3), endowed with the Boothby metric (in quaternionic coordinates), is a typical example. As to the locally conformal quaternion Kahler manifolds, examples can be obtained by acting freely with a discrete subgroup of GL(1, H) Sp(1) = COI (4) on HH" endowed with the Boothby metric (cf. [221], [222]). In both cases we have a Lee form w, locally defined by wlu = dfi and satisfying

d9 =wn9,

d0=0

'

(11.1)

where 9

Ha A Ha is the (global) Kahler 4-form. The properties (11.1) for 6 are also sufficient for a hyperhermitian or quaternion Hermitian metric to be 1.c.h.K. or l.c.q.K., respectively. The following result of P. Gauduchon, [104], now plays a crucial role:

Theorem 11.1 Let (N", [g], D) be a compact Weyl manifold, n > 3. i) There exists a metric go E [g], unique up to homotheties, whose associated Lee form wo is go-coclosed.

ii) If, in addition, (N", [g], D) is Einstein-Weyl, i.e. the symmetrized Ricci tensor of D is proportional to g, and wo is closed but not exact, then wo is parallel with respect to the Levi-Civita connection of go. Recalling that hyperKahler and quaternion Kahler metrics are Einstein (see e.g. (23]) the above theorem can be applied to our context. Precisely

Corollary 11.1 Let (M, g) be a compact locally conformal hyperKahler or locally conformal quaternion Kahler manifold and assume that no metric in the conformal class (g] of g is respectively hyperKdhler or quaternion Kdhler. Then there exists a go E [g] whose Lee form wo is V°-parallel.

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135

These (compact) manifolds are thus generalized Hopf ones. This is the property that makes the difference with respect to the complex case: we already encountered examples of compact l.c.K. manifolds with non-parallel Lee form (the Inoue surface with the Tricerri metric or the standard Hopf

surface blown up at one point, which has non zero Euler characteristic). By the above discussion, the following assumptions are not restrictive for compact manifolds and will always be made: (i) the fixed metric g makes w parallel (i.e. Ow = 0), (ii) I]w]] = 1.

As in the complex case, B = wO is the Lee vector field and ftQ the Kahler 2-form with respect to I. By usual computations (as in the complex case) one proves

Proposition 11.1 (cf. [221]) Let (M,g,II,72iI3) be a locally conformal hyperKahler manifold which is either compact or satisfying Vw = 0. Then the following formulas hold:

VB = 0,LBI,, = 0, LBg = 0, LBSla = 0, LB6 = 0

(11.2)

VI,, =

(11.3)

2{Id®woI, -Iaow -1Z®B+g®IQB} LI.BII = 0, LIQBIp = Iy, LI0Bg = 0

(11.4)

[B, IaB] = 0, [IaB, IQB] = IyB

(11.5)

(VXIaw)Y = 2 {I-w(Y)w(X) - w(Y)Iaw(X) - g(X, IuY)}

(11.6)

dIaw = 2OXIaw = S2a - w A Iaw

(11.7)

LI0BS2a = 0, LI,B110 = H-r, LI.B0 = 0,

(11.8)

where (a,,0, ry) = (1, 2,3) and cyclic permutations.

For a 1.c.q.K. manifold, the Weyl connection D does not preserve the compatible almost complex structures individually but only their 3-dimensional bundle H. In fact (cf. [226]),

DIa =

say,

Iµ,

(11.9)

where A, p = 1, 2, 3 and (aa,,) is a skew-symmetric matrix of local 1-forms. Accordingly, the above group of formulas have corresponding 1.c.q.K. ones, say (11.2)'-(11.8)'. E.g. (11.3)' is obtained by adding (DXIa)Y = F,, aa,,(X) I,,Y to the right-hand side of (11.3).

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We shall be interested in the properties of some canonically defined foliations F on l.c.h.K. and l.c.q.K. manifolds. We shall suppose, for simplicity,

that .F is regular in the sense that the leaf space is a Hausdorif manifold. However, in the weaker assumption of F having compact leaves, the leaf space is an orbifold (and this is the setting in the papers [221], [222] we report on in this section). Now let M be a compact l.c.q.K. manifold. As mentioned, we choose the metric g such that Vw = 0 and 11 w 11= 1. The vector fields B, I1B, I2B, I3B

span a 4-dimensional distribution V that, by formula (11.5)', is seen to be integrable (cf. [226]). Moreover Theorem 11.2 On a compact locally conformal quaternion Kahler (M, g, H), all the local quaternion Kahler metrics are Ricci-flat and M is therefore locally conformal locally hyperKdhler. Furthermore, the metric g is bundle-like with respect to V and projects to an Einstein metric with positive scalar cur-

vature on N = MID. Proof. The vanishing of the scalar curvature sD follows from Theorem 3 in [104] but it is interesting to give a direct proof in this context. The scalar curvatures s of g and sD of the quaternion Kahler g; on U; are related by D _ e_ l' r SIU _

Si

-

(4n-1)(2n-1)1 2

J

(cf. [23], p. 59). It follows that sD = const.; thus, if sD # 0, after derivation, Wi = dlog

(4n - 1)( 2n - 1) [slu -

2 1

Thus, since both w and s are global on M, w is exact, in contradiction with

our assumption. If s, = 0 on some U;, then s = slu, = (4n - 1)(2n - 1)/2, constant on M, hence s° = 0 for all U;. Then all local Kahler metrics are Ricci flat.

To see that g is bundle-like, as LBg = 0 by (11.2) we only have to compute L1. B9 on horizontal vector fields X, Y. By (11.3)' we derive

(GIaB9)(X+Y) = 9(OxIQB,Y) +9(X,VyIB) = ((VxIQ)B,Y +9(X, (VyI«)B) = E {a.,,(X)Iµw(Y) + {+

0.

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137

Hence, N is a Riemannian manifold and the projection M -> N is a Riemannian submersion. The Ricci tensors of M and N are related by 3

SN (X, Y) = SM(X, Y) + g(AxB, AyB) + > g(AxIaB, AyIaB) a=1

where Ax is the O'Neill tensor acting on vertical vector fields by AxV = hVX V, h being the horizontal projector (cf. [231, p. 244). We have AxB = 0 and, from (11.3)', AX I, B = - IQ X. On the other hand, taking into account a metrics are Ricci flat, on horizontal vector that the local quaternion Kahler fields we have SM = 2 11 g. Thus SN = (n + )gN and the proof is complete. 4

Remark 11.1 A consequence of the above result is that on compact 1.c.q.K. manifolds one may always assume the neighbourhoods U; to be simply connected and the local compatible almost complex structures to be integrable and parallel. However, the existence of a global hypercomplex structure on M is not implied (cf. examples below). Note that with respect to local parallel compatible Il, 12i 13 the matrix (aaµ) in formula (11.9) vanishes. Therefore, with this choice, the formulas (11.2)-(11.8) can be applied without modifications to compact l.c.q.K. manifolds.

A characterization of quaternion Kahler manifolds in the larger class of l.c.q.K. manifolds (here not necessarily non globally conformal) is given as follows.

Proposition 11.2 A l.c.q.K. manifold (M4n,g,H), n > 2, is quaternion Kahler if and only if through each point of it there exists a totally geodesic submanifold Q of real dimension 4h > 8, which is quaternion Kahler with respect to the structure induced on it by (g, H). Proof. The relation between the second fundamental forms of b; and b of a submanifold with respect to the metrics g= and g is easily seen to be

b', =b+g® B", where B" is the part of B normal to the submanifold. Let q E M and Q a quaternion Kahler submanifold as in the statement. Let j be the immersion of Q in M. Then j*de = 0, thus, cf. (11.1), j*wA j*O = 0. As rank(j*O) = 4h > 8, necessarily j*w = 0. It follows that the Lee vector field B is normal to Q, hence B = B". Further, j*w = 0 shows also that Q i Uj is a quaternionic submanifold of the quaternion Kahler manifold (Ui, gi'). Since

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138

quaternionic submanifolds of quaternion Kiihler manifolds are known to be totally geodesic (cf. e.g. [121]), this is the case for Q fl U; in U; with respect

to gi. Thus b = -g ® B on Q fl U;. But b = 0 from the assumption, thus B = 0 on Q fl U,. Since such a Q exists through any point q E M, it follows that B - 0 on M, i.e. g is a quaternion Kahler metric. The converse part is clear with Q = M. Let now M be a compact l.c.h.K. manifold, V the 4-dimensional foliation

on M defined above and Dl its orthogonal complement. The structure of M has been described in [226] and can be related to 3-Sasakian manifolds. The latter were extensively studied by C.P. Boyer & K. Galicki & B. Mann, [42]. Their considerations go back to the 1970's, starting with C. Udrigte, [265] who introduced them as "coquaternionic manifolds", and with Y.Y. Kuo, [166].

Proposition 11.3 (cf. [221]) (i) D is a Riemannian totally geodesic foliation. Its leaves, if compact, are complex Hopf surfaces admitting an integrable hypercomplex structure. In particular, they may be non primary. (ii) Dl is not integrable. Its integral manifolds are totally real and have maximal real dimension n - 1.

Proof. The proof of the first claim is very similar to the one of Theorem 5.1 so we omit it. If the leaves are compact, their structure can be deduced by Theorem 1 in [41]. They are in fact tangent to the Lee vector field B = wo, and hence carry a structure of hyperhermitian non hyperKiihler

4-dimensional manifold. As for (ii), the proof is very similar to that of Proposition 5.5. Here integrable hypercomplex structure is meant in the sense of G-structures, i.e. of the existence of a local quaternionic coordinate such that the differential of the change of coordinate belongs to H'. For further use we recall the following:

Theorem 11.3 (cf. [154]) A complex Hopf surface S admits an integrable hypercomplex structure if and only if S = (H - {0})/t where the discrete group r is conjugate in Gl(2, C) to any of the following subgroups G C H* C Gl(2, C): and r both cyclic generated by left multipli(i) G = Zm x rc with z

cation byam=e, m> 1, andcEC`.

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139

(ii) G = L x r,, where c E R` and L is one of the following: Dam, the dihedral group, m > 2, generated by the quaternion j and Pm = e+n; T24, the tetrahedral group generated by t;2 and 1 + (3j), S = e ; 048i ((3

the octahedral group generated by ( and 1 ((3 +1 (3j); 1120, the icosahedral group generated by E3, j,

LE4 - E + (E2

- E3)ij, E =

e21i.

(iii) G generated by Zm and cj, m > 3, c E R. (iv) G generated by D4m and cp2ni c E R* or by T24 and c(, c E R*. It follows from Theorem 11.2 that the leaf space N = MID of a compact l.c.h.K. M is an Einstein manifold with positive scalar curvature. Moreover, by (11.4) and (11.8), it is clear that although no single element of H projects

under p : M -+ N, this is the case for the bundle H itself and the Kiihler 4-form O. Thus (cf. [2261)

Proposition 11.4 Let M be a compact locally conformal hyperKiihler manifold of real dim > 12 with regular D. Then the leaf space N = MID inherits a structure of quaternion Kahler manifold with positive scalar curvature.

Proof. Denote by ON the projection of O on the leaf space. The vanishing of DON can be obtained by computing yO = F,a=1,2,3 V(1a Afta) on basic vector fields of the Riemannian submersion M --+ N. One obtains Ox11a(Y, Z) = g(Y, (OxI«)Z),

and for basic X, Y, Z the right hand side vanishes. The parallelism of ON then follows.

The result also holds for dim M = 8. Note that, as in the complex case, the fibration M -+ N can never be trivial. Remark 11.2 Any of the surfaces in Kato's list can actually occur as leaf of D. To see this, consider the standard hypercomplex Hopf manifold S41-1 x

S' = HH = (H" - {0})/I'2 and the diagonal action of any G in the list on the elements (ho, hl, ...h,,-,) of Hn - {0}. In this way, G acts on the fibers HPs-1, i.e. on the primary standard Hopf surface S3 x S'. of S4"-1 X S1 Note that, except in case (i) for m = 1, all fibers are non primary. To understand the above projection p : M -+ N better, we fix a complex structure J E H and consider the complex analytic foliation V,7 spanned by B and JB. From Theorem 6.3 and 1226) we then know that the leaf space Zj = M/Vj is a Kahler-Einstein manifold with positive scalar curvature. Let p be the projection M -+ Zj.

140

CHAPTER 11.

L.C. HYPERKAHLER MANIFOLDS

Remark 11.3 The above construction depends upon the choice of a complex structure J E H in the following way. For any J E H, Zi is analytically equivalent with the twistor space Z of N. (See [23] for the construction of the twistor space of a quaternionic Kiihler manifold.)

Proof. Fix q E Zj. Its counterimage on M is a complex torus on a well determined hypercomplex Hopf surface S, leaf of D. If r is the image of S on N, then q -- r defines a Riemannian submersion zr : Zj -' N and p = 7r o p. As the image of S under p is a sphere S2, 7r naturally realizes Zj as the total space of a S2-bundle over N. The fixed complex structure J on M projects on Zj under p but not on N under it. However, once a q in Zj is chosen, J defines a compatible complex structure Kq E H,. C End (TEN) by KqX = -7r.JXq, Xq being the horizontal lift at q of X E T,.N. This identifies the complex structure of Zj with that of the twistor space Z of N. We now consider the foliation B generated on M by the Lee vector field B and its orthogonal complement 131. The integrability of Bl is assured by

dw=0. Proposition 11.5 (cf. [221]) Bl is a totally geodesic Riemannian foliation. Its leaves have an induced structure of 3-Sasakian manifolds. In fact, each generalized Hopf structure (M, g, Ia) induces a Sasakian structure on the leaves. The three Killing vector fields IaB restricted to each leaf are related by the formulas g(IAB, IµB) = baN,,

[IAB, IB] = e' IB.

This is precisely the definition of a 3-Sasakian structure. Then, using Theorem 6.2 we obtain

Corollary 11.2 If B is regular, M is a flat S1-principal bundle over a 3Sasakian manifold P = M/B. The projection map is a Riemannian submersion.

On the other hand, by [42], 3-Sasakian manifolds are fiber bundles over quaternion Kahler orbifolds of positive scalar curvature with 3-dimensional homogeneous spherical space forms S3/r as fibers. Note that r is then one of the groups Z,,,, m > 1, D4r,,, m > 2, T24, 048, I12o appearing in Kato's list. Also, observe that all the compact hypercomplex Hopf surfaces classified by M. Kato are diffeomorphic to S3/I'-bundles over S1 ([154], p. 95). Hence,

by looking at the leaves of V and B, we see that the S3 /r still appear as fibering any 3-Sasakian manifold P = M/B which is the leaf space of a

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141

compact 1.c.h.K. manifold M having 8 regular. This provides the following structure theorem.

Theorem 11.4 (cf. [221]) (i) Let M be a compact locally conformal hyperKahler manifold and assume that the foliations D, V, B are regular. Then we have the following commutative diagram of fibre bundles (and Riemannian submersions).

The fibres of M N are Kato's integrable hypercomplex Hopf surfaces (S1 x S3)/G, not necessarily primary and not necessarily all homeomorphic. (ii) Conversely, given a quaternion Kahler manifold N of positive scalar curvature, there exists a commutative diagram as above with manifolds M', Z', P' respectively, locally conformal hyperKahler, Kahler-Einstein and 3-Sasakian and with fibers as described by r = G = Z2.

N associated to H -+ N. Proof. Consider the principal SO(3)-bundle P' Then P' has an associated 3-Sasakian structure and any flat principal Sl-

P' can be chosen to complete the diagram together with bundle M' the twistor space of N. Note that all arrows appearing in the diagram are canonical, except for M complex structure on M.

Remark 11.4

Z, which depends on the choice of a compatible

(i) This diagram holds also if dim(M) = 8. In this case

N is still Einstein by the above discussion. The integrability of the complex structure on its twistor space implies it is also self-dual (cf. [23]). Then recall that a 4-dimensional N is usually defined to be quaternionic Kahler if it is Einstein and self-dual.

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142

(ii) It is proved in [42] that in every dimension 4k - 5, k _> 3 there are infinitely many distinct homotopy types of complete inhomogeneous 3Sasakian manifolds. Thus, simply by making the product with S', we obtain infinitely many non-homotopically equivalent examples of compact l.c.h.K. manifolds. A first consequence of the diagram in Theorem 11.4 concerns cohomology.

Note first that the property Vw = 0 implies the vanishing X(M) = 0 of the Euler characteristic. Then, as in the similar, simpler diagram of the complex case (see Chapters 6, 7), applying the Gysin sequence twice in the upper triangle one finds the relations between the Betti numbers of M and Z:

bi(M) = bi(Z) + bi-1(Z) - bi-2(Z) - bi-3(Z) (0 < i < 2n - 1), b2. (M) = 2 [b2n-1(Z) - b2n-3(Z)]

On the other hand, since N has positive scalar curvature, both N and its twistor space Z have zero odd Betti numbers, cf. [23]. The Gysin sequence of the fibration Z - N then yields b2p(Z) = b2p(N) + b2p-2(N)

Together with the previous relations, this implies Theorem 11.5 (cf. [221]) Let M be a compact locally conformal hyperKdhler manifold with regular foliations B and D. Then the following relations hold: b2p(M) = b2p+1(M) = b2p(N) - b2p-4(N) (0 < 2p:5 2n - 2),

b2n(M) = 0.

Poincare duality gives the corresponding equalities for 2n + 2 < 2p < 4n. In particular b1 (M) = 1. Moreover, if n is even, M cannot carry any quaternion Kdhler metric. By applying S. Salamon's constraints on compact positive quaternion Kahler manifolds to the same diagram, it was proved in [100] that n

E k(n - k + 1)(n - 2k + 1)b2k(M) =0. k=1

Remark 11.5 (i) We obtain in particular b2p_4(N) < b2p(N) for 0 < 2p:5 2n - 2. Since any compact quaternion Kahler N with positive scalar curvature can be realized as the quaternion Kahler base of a compact 1.c.h.K. M

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143

(Theorem 11.4 (ii)), this implies, in the positive scalar curvature case, the Kraines-Bonan inequalities for Betti numbers of compact quaternion Kahler manifolds (cf. [231). (ii) b1 (M) = 1 is a much stronger restriction on the topology of compact 1.c.h.K. manifolds in the larger class of compact complex generalized Hopf manifolds. For the latter the only restriction is bl odd and the induced Hopf bundles over compact Riemann surfaces of genus g provide examples of generalized Hopf manifolds with b1 = 2g + 1 for any g, cf. Remark 6.3.

The properties b1 = 1 and b2n = 0 have the following consequences:

(i) Let (M, I1,12,13) be a compact hypercomplex manifold that admits a locally and non globally conformal hyperKahler metric. Then none of the compatible complex structures J = aIII + a212 + a313, al + az + a3 = 1, can support a Kahler metric. In particular, (M, 1,, 12,13) does not admit any hyperKahler metric.

(ii) Let M be a 4n-dimensional C° manifold that admits a locally and non globally conformal hyperKahler structure (1k, 12i 13, 9). Then, for n even, M cannot admit any quaternion Kahler structure and, for n odd, any quaternion Kahler structure of positive scalar curvature.

We shall now treat the homogeneous case. Unlike the complex case, where little is known, here a precise classification may be obtained. We call M a locally conformal hyperKahler homogeneous manifold if there exists

a Lie group which acts transitively and effectively on the left on M by hypercomplex isometries. We first prove

Theorem 11.6 (cf. [2211) On a compact locally conformal hyperKahler homogeneous manifold the foliations D, V and B are regular and in diagram (11.4) N, Z, P are homogeneous manifolds, compatible with the respective structures.

Proof. Let J E H be a compatible complex structure on M. Then (M, g, J) is a generalized Hopf homogeneous manifold and by Theorem 3.2 in [278] we

have the regularity of both the foliations V,7 and B. Therefore M projects on homogeneous manifolds Zj and P. In particular the projections of I,"B on P are regular Killing vector fields. Then Lemma 11.2 in [252] assures that the 3-dimensional foliation spanned by the projections of II B, 12B, I3B is regular. This, in turn, implies that N is a homogeneous manifold, thus V is regular on M.

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144

Starting with the Wolf classification of quaternion Kahler homogeneous manifolds, it is possible to classify 3-Sasakian homogeneous manifolds, cf. [42). This applies to our context and gives

Proposition 11.6 (cf. [2211) The class of compact locally conformal hyperKahler homogeneous manifolds coincides with that of flat principal Slbundles over one of the 3-Sasakian homogeneous manifolds:

S4n-1, RP4n-1

the flag manifolds SU(m)/S(U(m - 2) x U(1)),m > 3, SO(k)/(SO(k 4) x Sp(1)), k > 7, the exceptional spaces G2/Sp(1), F4/Sp(3), E6/SU(6), E7/Spin(12), E8/E7. The flat principal S1-bundles over P are characterized by having zero or torsion Chern class cl E H2(P; Z) and being classified by it. The integral cohomology group H2 of the 3-Sasakian homogeneous manifolds can be computed by looking at the long homotopy exact sequence ... -+ ir2(H) -+ 7r2 (G)

1r2(G/H) -+ irl(H) - 7r1 (G) -+ ...

for the 3-Sasakian homogeneous manifolds G/H listed above. Since 7r2(G) _ 0 for any compact Lie group G, one obtains the following isomorphisms (cf. [431):

H2 (

SU(m)

)

Z,

H2(RP4n-1) ^_' Z2

S(U(m - 2) x U(1) and H2(G/H) = 0 for all the other 3-Sasakian homogeneous manifolds. Hence

Corollary 11.3 (cf. [221)) Let M be a compact locally conformal hyperKahler homogeneous manifold. Then M is one of the following:

(i) a product (G/H) x S1, where G/H can be any of the 8-Sasakian homogeneous manifolds in the list

Son-1, RP4a-1, SU(m)/S(U(m - 2) x U(1)), m > 3, SO(k)/(SO(k - 4) x Sp(1)), k > 7, G2/Sp(1), F4/Sp(3), E6/SU(6), E7/Spin(12), E8/E7; (ii) the Mobius band, i.e. the unique non trivial principal S'-bundle over RP4n-1

.

For example, in dimension 8 one obtains only the following spaces: S7 x

S', RP7 X S', {SU(3)/S(U(1) x U(1))} x S' and the Mobius band over RP7. The first exceptional example appears in dimension 12: the trivial bundle {G2/Sp(1)} X S1 whose 3-Sasakian base is diffeomorphic to the Stiefel manifold V2 (R7) of the orthonormal 2-frames in R7.

We end this chapter with a result (whose proof is beyond the scope of this book) showing the close relation existing between l.c.q.K. and l.c.h.K. manifolds.

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145

Theorem 11.7 (cf. [2221) Let M be a compact l.c.q.K. manifold which is not quaternion Kahler. If the leaves of B are compact, then M admits a finite covering space carrying a structure of a l.c.q.K. manifold.

Chapter 12

Submanifolds 12.1

Fundamental tensors

Let (Mo", J, go) be a Hermitian manifold of complex dimension n (where J denotes its complex structure and go its Hermitian metric). Let 'P: Mm--+

Mo2n

be an immersion of a real m-dimensional CI manifold M' in Mon. To keep notation to a minimum, we do not distinguish between x and'P(x), or between X and W.X, etc., for any x E Mm, X E Tm(Mm). Let E('P) -, M"` be the normal bundle of the given immersion T. Set

PX = tan(JX) , FX = nor(JX)

t= for any X E T(Mm),

f = nor(J£)

E E(LY). Then

P E Hom(T(M'"))

,

F E Hom(T(Mm), E(WY))

t E Hom(E(T),T(Mm)) , f E Hom(E(WY)). Here tan, and norm are the natural projections associated with the direct sum decomposition

7T(Mo") = TT(M-) ® E(T).

(12.1)

for any x E Mm. Then the following identities hold: p2

= -I - tF

(12.2)

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148

FP + f F = 0

(12.3)

Pt+tf =0

(12.4)

f2 = -I - Ft

(12.5)

where I denotes the identity transformation. As in K. Yano & M. Kon, [296], p. 76-77, the identities (12.2)-(12.5) follow by applying J to JX = PX+FX

and J6 = t£ +

Note however that (12.2)-(12.5) hold for an arbitrary

ambient Hermitian manifold M02", while in [296] the identities (12.2)-(12.5) are stated under the assumption that go is a Kahlerian metric. We shall need the Gauss and Weingarten formulae:

V Y = OxY + h(X, Y)

0°x _ -A(X +V

(12.6)

(12.7)

for any X E T(Mm), E E(% F). Here V° , V are respectively the Levi-Civita connections of (Mo", go) and (Mm, g). Also h, AA and V1 are the second fundamental form (of W), the Weingarten operator (corresponding to the normal section ), and the normal connection (a connection in E(T) --> Mm). The connections V and V1 induce specific connections in the vector bundles Hom(T(Mm))

,

E(W)` ®T(Mm)

T(Mm)' ® E(%P) ,

Hom(E(W))

(for convenience all denoted by V). The corresponding covariant derivatives of P, F, t and f will be

(VxP)Y = VxPY - PVxY (VxF)Y = O1FY - FVxY

(VxtX = Vxtf - tvic (Oxf)( = OXff - fOzC for any X,Y E T(Mm),

E E(W). For the rest of this section, assume that (MO", J, go) is a 1.c.K. manifold (i.e. go is a 1.c.K. metric on (MO'", J)). Let V° be the Levi-Civita connection of (MO", go). Then, by (1.8) in Corollary 1.1,

OX JY + JOXY

+

2

{9o(Y)X - wo(Y)JX - go(X,Y)Ao - ilo(X,Y)Bo

(12.8)

12.1. FUNDAMENTAL TENSORS

149

for any X, Y E T(Mo"). Here wo, 9 are respectively the Lee and anti-Lee forms of (MM", J, go). Also Bo = woo and Ao = -JBo are the Lee and

anti-Lee fields on Mon. Raising of the indices is meant with respect to go. Finally 120 is the Kahler 2-form, i.e. SZo(X,Y) = go(X,JY). As M01 is l.c.K. its Kahler form Ho is not closed yet it satisfies (1.3) in Chapter 1 (it is closed if and only if go is a Kahlerian metric). Using (12.6)-(12.8) and a comparison between the tangential (respectively normal) components we derive the following expressions of the covariant derivatives VP, VF, Vt

and Vf : (VxP)Y = AFYX + t h(X, Y)

+2 {9(Y)X - w(Y)PX - g(X, Y) tan(Ao) -12(X, Y)B}

(12.9)

(VxF)Y = -h(X, PY) + f h(X, Y) - 2 {w(Y)FX + g(X, Y)A1 + 12(X, Y)B1}

(Vxt) =

(12.10)

PA£X

+2 {9o(C)X - w0(e)PX - flo(X,.)B}

(12.11)

(Vx f ) = -h(X, tC) - FAUX -2{wo(C)FX +12o(X,C)Bl}

(12.12)

for any X,Y E T(Mm) and any C E E(T). Compare with (1.6)-(1.9) in [296]. There go is assumed to be Kahlerian. As to the notations adopted in (12.9)-(12.12), we have

B = tan(Bo)

,

Bl = nor(Bo)

,

Al = nor(Ao)

w=W'W0 , 9=''80 12=T'120. As g = of*go it follows that 12(X, Y) = g(X, PY) for any X, Y E T(M'"). We shall need the equations of Gauss, Codazzi, and Ricci (cf. e.g. B.Y. Chen, [52]): go(Ro(X, Y)Z, W) = g(R(X, Y)Z, W) -go(h(X, W), h(Y, Z)) + go(h(Y, W), h(X, Z))

nor{Ro(X, Y)Z} = (V h)(Y, Z) - (Vyh)(X, Z) go(Ro(X,

77) =

-g([& A,,]X,Y)

CHAPTER 12. SUBMANIFOLDS

150

for any X, Y, Z E T(Mm) and £, 77 E E(LY). Here R, Rl and RD are the curvature tensor fields of V, Ol and V°, respectively. Also, we shall make use of the following form of the Gauss-Codazzi equations:

Ro(X, Y)Z = R(X, Y)Z - Ah(YZ)X + Ah(x,z)Y + (Oxh)(Y,Z) - (Vyh)(X,Z) for any X, Y, Z E T(Mm). Let KO be the curvature tensor field of the Weyl connection D° of the 1.c.K. manifold (Mo", J, go). A calculation (based on (1.5)) leads to

Ko(X,Y)Z = Ro(X,Y)Z

-

2 {Lo(X, Z)Y - Lo(Y, Z)X

+ go(X, Z)Lo(Y,

9o(Y, Z)Lo(X, (12.13)

4IIwoII2(X AY)Z

for any X, Y, Z E T (Mo"). Here Lo = V°wo + two ®wo.

Also the wedge product of two tangent vector fields X, Y on (Mo", go) is given by

(X A Y)Z = 9o(Y, Z)X

- 9o(X, Z)Y

Cf. e.g. [1261, vol. I, p. 73. If Mo" is a 1.c.K.o manifold (i.e. Ko = 0) then the Gauss-Codazi and Ricci equations may be written Ah(yZ)X - Ah(x Z)Y (Oxh)(Y, Z) + (Vyh)(X, Z)

R(X, Y)Z

+

1

2

{L(X, Z)Y

- L(Y, Z)X + g(X, Z)Lo(Y, )a

- g(Y, Z)Lo(X, )'} +

IIwoII2(X A Y)Z

9o(Rl(X,Y)e,,7) = 9((Af,Av)X,Y). The Ricci equation (12.15) is a consequence of Ro(X,Y)C =

2{Lo(X,)Y - Lo(Y,)X}

(12.14)

(12.15)

12.1. FUNDAMENTAL TENSORS

151

for any X, Y E T(Mm) and any C E E(W). In turn, this identity is obtained from (12.13) (with Ko = 0) for Z = . The (0, 2)-tensor field L in (12.14) is given by

L=W'Lo where raising of the indices is meant with respect to go. To derive the Gauss and Codazzi equations of a submanifold Mm of a 1.c.K.0 manifold we need to compare the tangential and normal components in (12.14). To this end, we need to compute tan{Lo(X, )O} and nor{Lo(X, )p} for any X E T(Mm). First, by using the Gauss and Weingarten formulae we get

tan{ OX Bo} = Vx B - ABiX

nor{OX Bo} = h(X, B) + V B1. Moreover

L(X, Y) = Lo(X, Y) = go(Lo(X, .)I, Y)

for any Y E T(Mm) so that tan{Lo(X, )p} = L(X, .)a

(12.16)

the raising of the indices is performed with respect to g. To write (12.16) in explicit form, note that

9(L(X, )a, Y) = L(X, Y) = (Voxwo)Y + 2w(X)w(Y)

= 9o(V%Bo, Y) + 2w(X)9(B,Y) Thus

L(X, )4 = tan{V Bo} + Iw(X)B, that is

L(X, ) _ VxB - AB-LX + 2w(X)B. Next, the calculation Lo(X,rl) (V Xwo)rl + 2W(X)WO(77)

go(OXBo,rl) + 1w(X)9o(Bo,77)

(12.17)

CHAPTER 12. SUBMANIFOLDS

152

furnishes

nor{Lo(X, )ti} = h(X, B) + OXB' + 1w(X)Bl.

(12.18)

As a consequence of (12.17)-(12.18) the Gauss-Codazzi equations (12.14) of a submanifold Mm in a l.c.K.0 manifold Mon may be written

R(X, Y)Z = Ah(yz)X - Ah(X Z)Y

+2 {L(X, Z)Y

- L(Y, Z)X

+g(X, Z)[VyB - ABlY + 2w(Y)B] -g(Y, Z)[VxB - ABIX + 1w(X)B]} +4IIwoII2(X AY)Z

(12.19)

(Oxh)(Y, Z) - (Vyh)(X, Z) =

2{g(X,Z)[h(Y,B)+V BJ-+ 1w(Y)Bl]

-g(Y, Z)[h(X, B) +VXB1 + Zw(X)Bl]}

(12.20)

for any X, Y, Z E T(M'). As a consequence of the Ricci equation (12.15) any totally geodesic submanifold of a 1.c.K.0 manifold has a flat normal connection. For further use, we set

L=Ow+2w®w so that (by the Gauss formula)

Lo(X,Y) = (X, Y) = L(X, Y) -wo(h(X,Y)) for any

X,Y E T(M).

12.2. COMPLEX AND CR SUBMANIFOLDS

12.2

153

Complex and CR submanifolds

The following notion is central for Chapter 12. Let (Mon, J, go) be a complex n-dimensional Hermitian manifold and %Y : M' -' M02n an immersion of a

m-dimensional manifold Mm in Mon. We call Mm a CR submanifold of (Mon, j, go) if Mm carries a CO° distribution V so that i) V is holomorphic (i.e. JJ(DV) = Dx for any x E Mm), and

ii) the orthogonal complement Dl with respect to g = Vgo of V in T(Mm) is anti-invariant (i.e. JJ(Dz) C E(W)x for any x E Mm). Cf. A. Bejancu, [161.

Let (Mm, D) be a CR submanifold of the Hermitian manifold Mon. Set p = dims Dx and q = dimR, Dy for any x E Mm so that 2p+q = m. If q = 0 then Mm is a complex submanifold, i.e. it is a complex manifold and T is a holomorphic immersion. If p = 0 then Mm is an anti-invariant submanifold (i.e. JJ(TT(Mm)) C E(WY)x for any x E Mm). A CR submanifold (Mm,D)

is proper if p # 0 and q # 0. Also (Mm, D) is generic if q = 2n - m (i.e. Jx(D2) = E(WY)x for any x E Mm). A submanifold Mm of the complex manifold (Mon, j) is totally real if

Tx(Mm) n JxT3(Mm) = {0}

for any x E Mm. Note that any anti-invariant submanifold is totally-real. Any orientable real hypersurface Men-1 of the Hermitian manifold Mon is a generic CR submanifold of Mon. Let M be a C°O manifold and T1,o(M) C T(M)®C a complex subbundle

(of the complexified tangent bundle) of complex rank p. Then Tip(M) is a CR structure (of CR dimension p) on M if T1,o(M) n T1,0(M) = {0} and

[T1,0(M),T1,0(1vI)] C T1,0(M).

Cf. e.g. S. Greenfield, [125]. An overbar denotes complex conjugation. A pair (M,Ti,o(M)) consisting of a C°O manifold and a CR structure (of CR dimension p) is a CR manifold (of CR dimension p). Let (M,T1,o(M)) and (N,T1,o(N)) be two CR manifolds and A : M - N a C°° map. Then A is a CR map if (d3A)T1,o(M)x C Ti,o(N)A(x)

for any x E M. A CR diffeomorphism is a C°° diffeomorphism and a CR map.

154

CHAPTER 12. SUBMANIFOLDS

For instance, let M C CN be a real submanifold and set Ti,o(M) = Tl'o(CN) n [T(M) ® C] where T""o(CN) denotes the holomorphic tangent bundle over CN. If

dimcTi,o(M)x = p = const. for any x E M, then Ti,o(M) is a CR structure (of CR dimension p) on M. The resulting CR manifold (M,TI,o(M)) is referred to as realized (in CN). By a result of D.E. Blair & BY. Chen, [31], any proper CR submanifold (Mm, D) of a Hermitian manifold Mo" is a CR manifold (of CR dimension p, where 2p = dimR Dx, for x E Mm). One may ask whether a given CR manifold (M,T1,o(M)) is realizable (i.e. CR diffeomorphic to a realized one). Only partial answers are available, even locally. For instance, by a classical result of A. Andreotti & C.D. Hill, [5], if (M,Ti,o(M)) is a real analytic CR manifold then it is locally realizable in CN, for some N. The same problem is largely unsolved in the C°° category. Counterexamples (i.e. examples of CR manifolds which are not realizable in any neighborhood of some point) exist, cf. e.g. L. Nirenberg, [200]. The positive embeddability results (cf. e.g. L. Boutet de Monvel, [39], M. Kuranishi, [167], T. Akahori, [3]) require additional assumptions (e.g. the strict pseudoconvexity of M). Realized CR manifolds have been widely studied, both from the point of view of analysis (e.g. the problem of holomorphic extension of CR functions on a CR manifold, cf. [34]) and geometry (e.g. the CR equivalence problem, cf. S.S. Chern & J. Moser, [65], H. Jacobowitz, [144], and (related) S. Webster's pseudohermitian geometry, cf. [286]-[288]).

The CR structure of the realized CR manifold is induced by the complex structure of the ambient space. If, additionally, a Hermitian metric is prescribed on the ambient space (as in A. Bejancu's approach, [15]) then it is natural, following the ideas in the theory of submanifolds in Riemannian manifolds, to attempt a classification of CR submanifolds with regard to the properties of their second fundamental forms. The monographs [296] and [15] illustrate the development in this direction. The interrelation between these viewpoints (the CR extension problem, the pseudoconformal invariants theory, etc., on the one hand, and the geometry of the second fundamental form in the presence of an additional Hermitian metric on the ambient space, on the other) has not yet been sufficiently investigated. Cf. S. Dragomir, [83], for a step toward filling in the gap between the two. Most of the results regarding the classification of CR submanifolds of a Hermitian manifold (Mo", J, go) are obtained under the assumption that

12.2. COMPLEX AND CR SUBMANIFOLDS

155

go is a Kahlerian metric, cf. [296]. Recently, the case where go is locally conformal Kahler has been taken under study (cf e.g. [46], [76], [216], [186], etc.). Here we report on results obtained mainly since the mid-1980s. There are a few remarkable differences between the geometry of CR submanifolds

in an ambient Kahlerian and 1.c.K. manifold. For instance, any complex submanifold of a Kahlerian manifold is known to be minimal. In the l.c.K. case (cf. Theorem 12.1 below) this is not the case unless the submanifold is tangent to the Lee field of the ambient space. To present this result, we shall need the following:

Lemma 12.1 Let Mm be a complex submanifold (m = 2p) of the l.c.K. manifold (Me", J, go). Then

h(X,JY) = J h(X, Y) - 2 {g(X,Y)A1 + S2(X,Y)Bl}

(12.21)

Aj X = J AtX - 2 (90(OX - wo(e)JX}

(12.22)

AtJX + J AfX = -ao(e)JX.

(12.23)

Proof. As JZT.,(Mm) = T2(Mm) for any x E Mm, it follows that

F=0 and JX = PX for any X E T(Mm). Next, let X E T(Mm) and l; E E(LY). Then 9(tC X) = 9o(Je, X) = -go(x, JX) = 0 because JX E T(Mm). We obtain t = 0 and Je = fl; for any C E E(!Y). Then (12.21)-(12.22) are direct consequences of (12.10)-(12.11), respectively. To prove the identity (12.23) we first use the symmetry of h and (12.21) to derive

h(JX,Y) = Jh(X,Y) -

2{g(X,Y)Al

- S2(X,Y)B1}.

(12.24)

Next, using (12.24) and

9(AfX,Y) = 9o(h(X,Y), we may conduct the following calculation:

9(A(JX,Y) = 2 {9(X,Y)9o(A', ) -go(h(X, Y),

- H(X,1')9o(Bl, )} {9(X,Y)eoW - H(X,Y)wo(e)}

CHAPTER 12. SUBMANIFOLDS

156

from which we get

A{JX = -AjeX -

2{9o(0X +wa(0JX).

(12.25)

Now (12.23) follows from (12.22) and (12.25).

At this point we may state (cf. I. Vaisman, [275)

Theorem 12.1 Let Mm be a complex submanifold of the l.c.K. manifold Mo". Then Mm is minimal if and only if the Lee field of M02" is tangent to MM.

Proof. Using (12.21), (12.24) and

J Al = B1, one may derive the following identity:

h(JX, JY) = -h(X, Y) - g(X,Y)B1.

(12.26)

Let {Ej} = {E,,,, JET } be a (local) orthonormal frame on M', m = 2p. Let H be the mean curvature vector of 41. Then by (12.26) m

H = 1m trace(h) = 1 Y h(EE, Ej) M E (h(EQ, Ea) + h(JEc,, JEa)) = m Ck=t

so that

H=-1Bj-

- m Ea g(E, E,,)Bl 1 (12.27)

2

and Theorem 12.1 is proved. The general philosophy emerging so far is that one should attempt to a classification of CR submanifolds Mm of a l.c.K. manifold M02" not only with regard to properties of the second fundamental form (of the given immersion T of Mm in M02") but also with regard to the position of 'I (M'") with respect

to the Lee field of Mo. Natural limitations may occur. For instance, we have (cf. N. Papaghiuc, [223]):

Proposition 12.1 Let M be a m-dimensional (m > 2) CR submanifold of a Vaisman manifold M02". If the anti-Lee field is normal to M then M is an anti-invariant submanifold of M02" (and m < n). Consequently, a Vaisman

12.2. COMPLEX AND CR SUBMANIFOLDS

157

manifold admits no proper CR submanifolds so that A = 0. In particular, there are no proper CR submanifolds of a Vaisman manifold with B0 E D1. Also, there are no complex submanifolds of a Vaisman manifold normal to the Lee field.

Proof. Let N10n be a g.H. manifold. As a consequence of D°Bo = 0 and of (12.8) we have VXAo = a{Ilwoll2JX +wo(X)Ao - Oo(X)Bo}

for any X E T(Mon). Let (M, V) be a CR submanifold of M02. Let X,Y E T(M). If A = 0 then W`Bo = 0 and we may conduct the calculation

0 = go([X, Y1, Ao) = go(V Y, Ao) - go(V X, Ao)

= -go(Y,DoxAo)+go(X,V Ao) llwoll2f (X, Y)

that is 0. Consequently J(T(M)) C E(P), Q.E.D. Clearly, if Bo E D1 then Ao = -JBo E E(LP) and the first statement in Proposition 12.1 applies. Let M be a complex submanifold of M02'. If B = 0 then A = 0 (because of JE(WY) = E(LY)) and, by the first statement in Proposition 12.1, M should be totally real, a contradiction. Next, we examine the effect of positive holomorphic bisectional curvature on the properties of the normal bundle (of the given holomorphic immersion). Let Rizo be the holomorphic bisectional curvature of (Mon, j, go), cf. G.B. Rizza, [235] (where the concept of holomorphic bisectional curvature was first introduced). Cf. also (1131. Precisely, let a, o' be two holomorphic 2-planes tangent to Mon at a point x. Let u E or and v E o' so that lull = 1, Ilvll = 1. Then, by definition

Rizo(a, o') = (Ro,x(v, Jxv)Jxu, u)

where (,) = go,x. The definition of Rizo(o, o') does not depend upon the choice of unit vectors u, v in a, a', respectively. Cf. [236]. We may state the following (cf. (861):

Theorem 12.2 Let M2p be a complex submanifold of the I.C.K. manifold Mpn. If Mon has positive holomorphic bisectional curvature then the normal bundle of the given immersion of M2p in Mon admits no parallel sections.

CHAPTER 12. SUBMANIFOLDS

158

Proof. The proof is by contradiction. Let l; be a parallel normal vector field

(i.e. V le = 0). Then Rl (X, Y)e = 0 for any X, Y E X(M2P). Set rj = Jt; in the Ricci equation. We obtain Ro is the Riemann-Christoffel 4-tensor of (Mon, go). That is,

Ro(U, V, Z, W) = go(Ro(U, V)Z, W)

for any U, V, Z, W E X(Mo"). As a direct consequence of (12.21)-(12.23) in Lemma 12.1 one may derive the following:

Lemma 12.2 Let M2p be a complex submanifold of the l.c.K. manifold Mon. Then [At, Ajt] = -2JT42 (12.29) for any l; E E(LY), where

TT=At +2wo(e)I and I denotes the identity transformation. Using Lemma 12.2 we may rewrite (12.28) as Ro (X, Y, C, JC) = 2g(JT42X, Y).

Set Y = JX. Let a, a' be the holomorphic 2-planes spanned by {u, Jxu} and {v, Jxv}, respectively, where u = (X/II X II)= and v = (/IIEII)z As TT is self adjoint, we obtain IIXII2IIeII2Rizo(a,a') = -2I[T4I[i < 0.

a contradiction.

Compare with B.Y. Chen & H. Lue, [58]. A deeper investigation of the properties of the normal bundle of a complex submanifold of a I.c.K. manifold

is undertaken in Chapter 15.

12.3

Anti-invariant submanifolds

Let Mm be an anti-invariant (i.e. JXTX(Mm) C E(WY)x, for any x E Mm) submanifold of the Hermitian manifold Mo". Then P = 0. If additionally Mm is generic (i.e. m = n) then f = 0. By Proposition 1.3 in [296], p. 82, an n-dimensional anti-invariant submanifold M" of a Kahlerian manifold M02" has a flat normal connection if and only if (the induced metric of) Mn is flat. As to the 1.c.K. case, we establish the following:

12.3. ANTI-INVARIANT SUBMANIFOLDS

159

Proposition 12.2 Let M' be an n-dimensional anti-invariant submanifold of the 1.c.K. manifold Mon. Then Mn has a flat normal connection if and only if

R(X, Y)Z = 2 {L(X, Z)Y - L(Y, Z)X+ + g(X, Z)L(Y, )p - g(Y, Z)L(X, .)p} + 4 IIwII2(X n Y)Z

(12.30)

for any X, Y, Z E T(M").

Proof. Let D° be the Weyl connection of Mon. Since D°J = 0 it follows that

Ko(X,Y)JZ = JKo(X,Y)Z.

(12.31)

Let us substitute KO from (12.13) into (12.31) to obtain the identity

Ro(X,Y)JZ = JRo(X,Y)Z +2 {Lo(X, JZ)Y - Lo(Y, JZ)X -Lo(X, Z)JY + Lo(Y, Z)JX +go(X, JZ)Lo(Y, )p - go(Y, JZ)Lo(X, -go(X, Z)J Lo(Y, .)0 + go(Y, Z)J Lo(X, .)a}

+4IIwo112 ((X A Y)JZ - J(X A Y)Z).

(12.32)

Let X, Y, Z, W E T(M") (so that JZ, JW E E(LY)). Take the inner product of (12.32) with JW and use the Ricci and Gauss equations to yield go(R1(X,Y)JZ, JW) - g([Aiz, Arw]X, Y) = g(R(X, Y)Z, W) - go(h(X, W), h(Y, Z)) + go(h(Y, W), h(X, Z))

+2 {L(Y, Z)g(X, W)

- L(X, Z)g(Y, W)

+L(X, W)g(Y, Z) - L(Y, W)g(X, Z)} +4IIWoII2 (g(X, Z)g(Y, W) - g(Y, Z)g(X, W)). We need the following:

(12.33)

CHAPTER 12. SUBMANIFOLDS

160

Lemma 12.3 Let M" be an n-dimensional anti-invariant submanifold of the l.c.K. manifold Mo". Then

AjyX = -Jh(X, Y) - {O(Y)X +g(X,Y)JBI} 2

(12.34)

and

g([Arz, A.w]X, Y) = go(h(X, W), h(Y, Z)) - go(h(X, Z), h(Y, W))

+2 {wo(h(X, W))g(Y, Z) - wo(h(X, Z))g(Y, W) +wo(h(Y, Z))g(X, W) - wo(h(Y, W))g(X, Z) +6(Z)Sto(Y, h(X, W)) - 9(W)S2o(Y, h(X, Z))

+o(W)clo(X, h(Y, Z)) - o(Z)S2o(X, h(Y, W))}

+4 {o(Z)o(X)g(Y,W) -O(W)O(X)g(Y,Z) +O(W)O(Y)g(X, Z) - o(Z)o(Y)g(X, W) +IIB1112 [g(X, W)g(Y, Z) - g(X, Z)g(Y,

W)])

(12.35)

for any X, Y, Z, W E T(M" ).

Proof. It is easily seen that (12.34) follows from (12.9) because of P = 0. Also (12.35) follows by substitution from (12.34).

Let us go back to the proof of Proposition 12.2. We substitute from (12.35) into (12.33) and use the identity S2o(Y, h(X,

Z)) = SZo(Z, h(X,Y)) + 2 {o(Y)g(X, Z) - o(Z)g(X,Y)}

to obtain

-J Rl (X, Y)JZ = R(X, Y)Z + 2 {L(Y, Z)X - L(X, Z)Y +g(Y, Z)L(X,

g(X, Z)L(Y, )1 }

+4 11wII2{g(X, Z)Y - g(Y, Z)X }

for any X, Y, Z E T(M'). Thus Rl = 0 if and only if (12.30) holds. Note that S. Ianu§ & K. Matsumoto & L. Ornea prove Theorem 5.3 in (133), p. 129, under the assumption that the complex hypersurface Men-2

12.3. ANTI-IVVARIANT SUBMANIFOLDS

161

admits recurrent cross-sections l; in the normal bundle, i.e. VX£ = a(X)l; for some 1-form a on M2i-2. If this is the case, then the curvature 2-form R1 of V-L satisfies Rl(X,Y)1; = (da)(X,Y)e. We slightly generalize this situation by calling Mn` a submanifold of recurrent normal curvature if Rl(X,Y)1; = O(X,Y)C

(12.36)

for any X, Y E T(Mm), 1; E E(LY), and for some 2-form a on M'n. Then (cf. [87])

Theorem 12.3 Let 11TH be an n-dimensional anti-invariant submanifold of recurrent normal curvature of the 1. c. K. manifold Mon. Then Mn has a flat normal connection. Proof. Using (12.30) in Proposition 12.2 and assumption (12.36) we derive

R(X,Y)Z = e(X,Y)Z + 2 {L(X, Z)Y - L(Y, Z)X +g(X, Z)L(Y, )a - g(Y, Z)L(X, )d} +4IIwII2(X A Y)Z.

Next, a suitable contraction of indices gives the Ricci form Ric of (M', g), that is

Ric(X,Y) = -e(X,Y) -

n

2

2L(X,Y) + {n

2

2IIwII2

- div(B)}g(X,Y). (12.37)

Since V is torsion free, L is symmetric. But Ric is symmetric, so that (12.37) yields e = 0 and our Theorem 12.3 is proved. We recall that a 1.c. K. manifold (Mon, J, go) is quasi-Einstein, a terminology introduced in [115], if the Ricci tensor Rico of (Mr, go) is given by Rico = ago + bran ® wo for some COO functions a, b on Mon.

Quasi-Einstein manifolds are abundant in l.c.K. geometry. For instance the complex Hopf manifold CHa , carrying the Boothby metric go, is quasiEinstein. Also (by Theorem 2 in [75], p. 200) any totally umbilical complex submanifold of (CHa , go) is quasi-Einstein. If MI is a submanifold of a l.c.K. manifold Man then, by slightly generalizing the situation above, we call Mm quasi-Einstein if the Ricci tensor of (Mm, g) is given by Ric = ag + bw ®w

(12.38)

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162

for some a, b E C' (M')

-

Assume Sing(w) = 0. Then Mm carries the canonical foliation F whose leaves are the maximal connected integral manifolds of the Pfaffian equation

w=0. Let Mon be a Vaisman manifold (i.e. Vowo = 0). We set [Iwo[[ = 2c, c E [0, +oo). Let Mm be a submanifold of the g.H. manifold Mpn. Then, by the Gauss formula (Vxw)Y = wo(h(X,Y)) (12.39)

for any X, Y E T(Mm). If additionally Mm is totally geodesic (h = 0) then

w is parallel (so that liwil = const.). Then either Sing(w) = 0 or M' is normal to the Lee field of n101n

Theorem 12.4 Let M" be a connected complete totally geodesic anti-invariant submanifold (of real dimension n > 2) of the Vaisman manifold Mon. Suppose Mn has a flat normal connection. i) If Mn is a simply connected real surface (n = 2) tangent to the Lee field (B-L = 0) of the ambient g.H. manifold M4 then it is isometric to R2 with the flat Euclidean metric. ii) If n > 2, Bl # 0 and Mn is orientable then Mn is normal to the Lee field and its universal covering manifold is isometric to the sphere S"(1/c).

iii) If n = 2, B-L # 0 and M2 is orientable then M2 is isometric to S2(1/c). iv) If n > 2 and Bl = 0 then Mn is quasi-Einstein and its Ricci tensor is positive semi-definite and degenerate only along the distribution generated by the Lee field. Moreover, any leaf of the canonical foliation F of Mn is a totally geodesic real hypersurface of Mn and its universal covering manifold is isometric to Si-1(1/c).

Throughout S(r) denotes the sphere of radius r > 0 and center the origin of R'"+i (carrying the standard Riemannian metric). Proof. Let Mn be an n-dimensional anti-invariant submanifold of the g.H. manifold Mon. Assume Rl = 0. Then R is expressed by (12.30). If additionally Mn is totally geodesic then (12.30) becomes R(X, Y)Z = 4 { [w(X)Y - w(Y)X]w(Z) +[g(X, Z)w(Y) - g(Y, Z)w(X)]B} +c2{g(Y, Z)X - g(X, Z)Y}

(12.40)

12.3. ANTI-INVARIANT SUBMANIFOLDS

163

for any X, Y, Z E T(M). Then, a suitable contraction of indices leads to

Ric(X,Y)

41

(n - 1)c2 - 4IIwII2 } g(X,Y) -

-n 4

2w(X)w(Y).

(12.41)

Now let M2 be a totally geodesic anti-invariant surface in a g.H. manifold M4 of complex dimension 2. By (12.41) one has Ric = 1 JIB1II2g.

Therefore, if M2 is tangent to Bo, then it is Ricci flat, and thus flat for surfaces the two conceptss are known to coincide. Suppose from now on that n > 2. If X is tangent to M", then (12.41) leads to Ric(X, X) _

-n

2w(X)2+4{(n-1)IIBIII2+(n-2)IIwII2}IIXII2. (12.42)

4

Since IIwII = const., either w = 0 and then (12.40) shows that Mn is a Riemannian manifold of constant sectional curvature c2 > 0, or w is nowhere vanishing. The last step in the proof of ii) consists of showing that the second possibility does not actually occur. To this end, suppose w i4 0. Let x E Mn and X E TT(Mn). There exists a unique Y E TT(Mn) perpendicular to Bz such that X = Y + AB-, for some A E R. Substitution into (12.42) leads to

Ric(X, X) = 4,\2(n -1)IIB1II2IIWII2

+4 [(n - 1)11 B' 112 + (n - 2)IIwI12] 11Y112,

that is, Ric(X,X) > 0. Also Ric(X,X) = 0 if and only if Y = 0 and A = 0, since B1 # 0. Thus Ric is positive-definite. By a result of [195], Mn follows to be compact. Since w is parallel, it is (by (2.12.4) in [106], p. 78) harmonic. Consequently, the first Betti number b1(M") of Mn is > 1. This is a contradiction, since by a result of S. Bochner (cf. (106], Theorem 3.2.1, p. 87) b1(M") = 0, Q.E.D. Let us prove iii). If n = 2, B-L 54 0, then by (12.41) it follows that M2 is a compact Einstein surface (with b1 (M2) = 0). But w is harmonic, hence w = 0. Consequently, by (12.40) and a result of [248], 7r1(M2) = 0. At this point Theorem 7.10 of [162], vol. I, p. 265, yields iii). Moreover, if

n > 2, B1 = 0, then Ric = n 4 2 [IIwII2g - w ®w,

CHAPTER 12. SUBMANIFOLDS

164

M' is quasi-Einstein. Also, if X = Y + AB, g(Y, B) = 0, then Ric(X, X) = 0 if and only if Y = 0. To prove the last part of the statement iv), let M"-1 be a leaf of F. As Mo" is non-Kiihler and B1 = 0, the induced form w has no singular points. Then U 2c B i.e.

is a unit normal on Mn-1. Let V', h' be the induced connection and the second fundamental form of Mn-1 in M' respectively. The Gauss formula

VxY = V' Y + h'(X,Y) and OXw = 0 yield h' = 0, i.e.

MI-1 is to-

tally geodesic. Then, on the one hand, the completeness of M" implies the completeness of MI-I. On the other, our (12.40) combined with the Gauss equation (e.g. (2.6) in (52], p. 45) of M"-1 in M" shows that M'-1 is a Riemannian manifold of constant sectional curvature c2 > 0.

12.4

Examples

Let Mo" be a l.c.K. manifold and Do its Weyl connection. Let Mm be a submanifold of My". Let us set DXY = tan{DXY} , DXY = nor{DXY} for any X, Y E T (M'") , E E(% F). Clearly, D is a linear connection on Mm, while D-L is a connection in the normal bundle E(l). Let

h'(X, Y) = nor{DXY}

,

At'

tan{DXY}

for any X, Y E T(Mm), E E(LY). Then h', A' are C°O(M)-bilinear. Let {90,i}iE1 be the local Kahler metrics of M. Let gi ='T''9o,i be the metric induced on Mm by go,i. Then D is the Levi-Civita connection of gi. Therefore, (local) facts concerning the geometry of submanifolds in Kiihler manifolds (formulated in terms of V, h, A, V1) may be rephrased in terms of D, h', a', D1, as is often done in the following. The following identities hold:

DXY = DxY + h'(X, Y), Do = -at'X + DXY

Dg+w®g=0 go(h'(X,Y),.) = 9(A'X,Y) Using the Gauss and Weingarten formulae, we obtain

DXY =VXY - 2{w(X)Y+w(Y)X -g(X,Y)B}

(12.43)

12.4. EXAMPLES

165

h'(X, Y) = h(X, Y) + Ig(X,Y)Bl A£X =

(12.44)

2wo(l;)X

(12.45)

DX = X - 2w(X)t;

(12.46)

for any X,Y E T(Mm) and any £ E E(1Y). This is the proper place to furnish to the reader a few examples of submanifolds of 1.c.K. manifolds (cf. [76], p. 194-196). 1) Let j : C' -+ C" be the natural injection, i.e. (zl, .. , z'") (z', , z'", 0, , 0), m < n. It induces an embedding j : CHa -+ CH, \. Its first fundamental form g = j'go coincides with the Boothby metric of CHa . Let D° be the Weyl connection of CH, \. For any X, Y E T(CHa ), DXY is tangent to CHI. This, our (1.5), and the observation that j'wo is precisely the Lee form of CHa , show that CHa --+ CHa is totally umbilical. Clearly, CH' is invariant. Then CHa -+ CHI is a complex Hopf surface embedded

in CH' , n > 2. The torus T2 = CHa is a totally umbilical real surface in CH,\ . By a result in [75] (cf. Theorem 12.19 below), the only totally umbilical invariant submanifolds of zero scalar curvature r are the totally geodesic (flat) real surfaces; yet for T2 -+ CHa we have r = 211H112 3k 0.

2) In general, if i : M -+ (Cn - {0}, IxI-16,j) is a submanifold, then 9r: M -+ CHa, W = it o i, is an isometric immersion (where M is endowed

with the R.iemannian metric induced via i). By (12.44), if M -+ (Cn {0}, 6ij) is totally geodesic, then 'Y is totally umbilical. Indeed, for any VCn

is the Riemannian X,Y E T(Cn) one has VO,x7r.Y = 7r.VcnY where connection of IxI-28ij (cf. [162], vol. I, p. 161). Thus hi, = 7r.h where hq,, h

are respectively the second fundamental forms of 'P and of M -+ (C" {0}, IxI-26i,). On the other hand h' = h + (1/2)g 0 B'. Therefore H' = exp(f)(H+ (1/2)Bl) , f = log Ix12. Finally we obtain h4 = hO Hq', Q.E.D. Analogously, one may show that if M -+ (C" - {0}, Sij) is totally umbilical, then 'Y is totally umbilical. 3) Let Mp" be a l.c.K. manifold and Fo the canonical foliation of M02" whose leaves are the maximal connected integral manifolds of the Pfaffian equation wo = 0. Each leaf of F0 is a totally geodesic submanifold of Mon, cf. [269], p. 270. As a consequence of our Theorem 12.22, the totally geodesic real surfaces M2 in CHn may be classified; M2 is a space-form M2(k), where either k = 2, when M2(k) is normal to the Lee field B0 of CH', or k = 0, when M2(k) is tangent to Bo. Since IIw[[ = const., no other situation occurs.

CHAPTER 12. SUBMANIFOLDS

166

4) Let M -+ (C" - {0}, bij) be minimal. Then T : M -> CHa is subject to H = -(1/2)B1. If WY' are the components of IF with respect to some coordinate system (U, x'), 1 < i < 2n, on CH", then one may derive (cf. Theorem 15.7 below) OT' = mH' - B(WY') + (m/2)Bo('Y') (where

m = dim(M)). Thus, if M is a real surface (m = 2) then the 'Y' are harmonic. 5) Cf. [133], we recall the following example. Let M02" be a compact

generalized Hopf manifold. Let U = ua, u = IIwoII-1wo, and V = JU. If {U, V) determine a regular foliation F (i.e..F = D1(D D2i cf. Chapter 5) on M0" then p : Mo" -+ M02"/.F is a locally trivial fiber bundle over a compact

Kahlerian manifold. Then for each complex hypersurface N in M0n/.F, p-1(N) is a complex hypersurface of M0" and both UV are tangent to p' (N). See also [64], remark 8.4, p. 303. 6) Let

i : M -+ C(n+k)/2 M = Sml (rl) x ... X Smk (rk) , n = ml + ... + Mk

where mi, 1 < i < k, are odd. Then let W : M -' CHA"+k)/2 , Y = 7r o i, By where 7r as usual denotes the projection C("+2)/2 - {0} -4 CHa"+k)/2.

Example 7.2 in [296], p. 99, M -' CHa"+k)/2 is CR, verifies VXH = h(X, B) for any X E T(M), and has a flat normal connection.

7) If A = e2 we recall the diffeomorphism F : CHa . S1(1/ir) X S2n-1 is a submanifold, then M = S1(1/7r) x N (cf. Chapter 3). If N gives rise to a submanifold of CHa , in a natural way. Let Ek C C°O(S'n) be the eigenspace (of the Laplacian on the sphere S'") corresponding to the eigenvalue -k(m + k - 1). Cf. [22], p. 162, we S2n-1

have

dimll.Ek=(m+k12 I{2+mk 1}. Therefore, by a theorem in T. Takahashi, [2255], we may choose p > 0, a > 0 and an orthonormal basis of E2 C COO(SS), say {W1, , W20} to produce

a minimal immersion cp = (apt, ... ,co) (p2: S5 - S19(p). Let M20 be a compact g.H.o manifold of complex dimension 10. We may norm the Hermitian metric of M20 in such a way that IIwo]I = 2/p. By Theorem 3.8 in [269], p. 277, the universal covering space of M20 is 00 - {0} with the M20 be the natural projection. metric p2izl-2bijdz'dzi. Let 7r : 00 - {0} Finally, we get the isometric immersion WY : S'(1/7r) X S5 - M20 given by IF = 7r o F o (cp x ISl(1/,r)).

12.5. DISTRIBUTIONS ON SUBMANIFOLDS

167

8) Let j : Sl (1 / f) x S1(1/ J) - S3 be the Clifford torus, and W = i o j,

where i:S3-+R4;:: C2. Then W:Sl(1/f)xSl(1/v)-'CHais an isometric immersion whose mean curvature vector is given by H = -7r.cp. Indeed, by a theorem in [255], since j is minimal, Ocp = -2w. But zc' is twice the mean curvature vector of

Sl(1/f) X Sl(1/v)

- {0}. 9) Let CHa - CPn-1 be the Hopf fibration. Let Qn-2 : (zo)2 + + (zn-1)2 = 0 be a hyperquadric of CPn-1. Then is a complex C2

7r-1(Qn-2)

hypersurface in CH, \.

Further examples are considered in Theorem 12.10 (giving an isometric CH' with B = 0), in Theorem 12.11 (giving a totally immersion in CHI), and in umbilical immersion of the real Hopf manifold Theorem 12.12, indicating examples of totally umbilical real hypersurfaces, respectively of totally geodesic complex hypersurfaces, in a complex sphere Qn_1. Examples of real hypersurfaces in a complex Inoue surface SM are considered in Section 14.5. Cf. also Section 16.1 for the example S2n-1

RH,2n-1

Vmi,...,mk ' CHan+k+l)/2

of a generic CR submanifold with a flat normal connection and a parallel second fundamental form.

12.5

Distributions on submanifolds

Let (Mpn,J,go) be a 1.c.K. manifold. Let (Mm,V) be a CR submanifold of Mon. By a result of B.Y. Chen, [54], the anti-invariant distribution Dl of a CR submanifold of a Kahler manifold is integrable. This is still true for ambient 1.c.K. manifolds (so that any CR submanifold of a l.c.K. manifold comes naturally equipped with a totally real foliation). We have (cf. D.E. Blair & B.Y. Chen, [31]) Theorem 12.5 Let (Mm,D) be a CR submanifold of the l.c.K. manifold Mon. Then the anti-invariant distribution D1 of MI is integrable. Proof. Let X E V and Z, W E Dl. Then 1lo(X, Z) = 0 and 1Zo(Z, W) = 0. Consequently (Q0 A wo)(X, Z, W) = 0 and hence (by (1.3))

0 = 3(dIZo)(X, Z, W) = X (1 o(Z, W )) + Z(Ho(W, X )) + W (flo(X, Z))

co([X, Z],W) - S1o([W,X], Z) - fzo([Z,W],X)

-g([Z, W], M.

CHAPTER 12. SUBMANIFOLDS

168

Therefore [Z, W] E Dl, Q.E.D. Let (Mm, D) be a CR submanifold of a Hermitian manifold M02 n. By slightly generalizing a concept of A. Bejancu & M. Kon & K. Yano, [19], we call Mm mixed totally geodesic if t h(X, Y) = 0

for any X E V and any Y E Dl. If Mm is generic (f = 0) then our notion (of mixed totally geodesic CR submanifold) coincides with the one in [19] (they request h(D,V1) = 0). We have (cf. S. Dragomir, [76], p. 182)

Theorem 12.6 Let Mm be a CR submanifold of the l.c.K. manifold Mo". Assume Mm to be normal to the Lee field of Man. Then Mm is mixed totally geodesic if and only if each leaf of the anti-invariant distribution Dl is a totally geodesic submanifold of Mm. Proof. We need the following:

Lemma 12.4 For any CR submanifold (Mm, V) of a Hermitian manifold we have

PT(Mm) = D

,

PD1 = 0

t E(T) = D1-.

Proof. Let X E T(Mm) and Y E Dl. Then g(PX, Y) = go(JX,Y) = -go(X, JY) = 0

so that PT(Mm) C D. Let X E D. Then JX E V, i.e. JX = Y for some Y E D. Thus X = P(-Y), that is, D C PT(Mm). Let Y E Dl. Then JY E E(IF) so that PY = tan(JY) = 0. Let l; E E(') and X E D. Then g(tc, X) = go(A, X) = -go(c, JX) = 0

so that t is D1-valued. Let Y E Dl. Then JY E E(WY). Set Then ti; = tan(Jl;) = Y, Q.E.D.

-JY.

Lemma 12.5 Let Mm be a CR submanifold of the l.c.K. manifold Mon. Let X E D and Y, Z E D1. Then the following identity holds:

-g(tVyZ,JX) = g(t h(X, Y), Z) + 1O(X)g(Y,Z) where t : T(Mm) -* D is the natural projection.

(12.47)

12.5. DISTRIBUTIONS ON SUBMANIFOLDS

169

Proof. As PY = 0 and Sl(Y, Z) = 0, the identity (12.9) becomes

-PVyZ = AFZY + t h(Y, Z) + 2 {9(Z)Y - g(Y, Z)A}. Take the inner product with X and use the fact that t is Dl-valued. We get

g(PVyZ,X) = -go (h(X, Y), FZ) + 1O(X)g(Y,Z) Finally

g(PW,X) =g(PtW,X) = -g(tW,JX) for any W E T(Mm) and the proof of Lemma 12.5 is complete. At this point we may prove Theorem 12.6. Assume B = 0. Then 0(D) _ 0 and (12.47) in Lemma 12.5 becomes

g(t OyZ, JX) = -g(t h(X,Y), Z) for any X E D and any Y, Z E D. The proof of Theorem 12.6 now follows by observing that, for each leaf S of Dl, the second fundamental form hs of S in Mm is given by

hs(Y,Z) = tVyZ for any Y, Z E T(S) = Dl as the normal bundle of S --' Mm is the portion of S over D. Remark 12.1 Note that Theorem 12.6 (cf. also Theorem 3 in [76], p. 182) extends Corollary 4.1 in [2961, p. 92, (from the case of an ambient Kahlerian manifold to the case of an ambient 1.c.K. manifold). An inspection of (12.47) shows that Theorem 12.6 admits a natural improvement, as follows. Indeed, let Mm be a CR submanifold of the l.c.K. manifold M02° (no assumption on

the position of Mm with respect to B0 is made). If Mm is mixed totally geodesic then (12.47) yields

t OyZ = -g(Y, Z)t B for any Y, Z E Dl, so that we may state the following:

Theorem 12.7 Let MI be a CR submanifold of the l.c.K. manifold M02" Then Mm is mixed totally geodesic if and only if each leaf S of Dl is a totally umbilical submanifold of Mm and its mean curvature vector Hs is given by Hs = -q t B (q = dimR, Dl).

170

CHAPTER 12. SUBMANIFOLDS

Let M' be a CR submanifold of a Hermitian manifold. Then M' is referred to as D-geodesic if h(X, Y) = 0 for any X, Y E V (cf. the terminology adopted in [191).

Theorem 12.8 Let M' be a proper CR submanifold of the l.c.K. manifold Mon.

i) The holomorphic distribution D of M'n is integrable and its leaves are totally geodesic in M' if and only if for any X, Y E V and any Z E Dl one has

go(h(X, Y), JZ) + 1 g(X, Y)O(Z) = 0

(12.48)

and any leaf of V is tangent to B. Moreover, if (12.48) holds and M' is generic then all leaves of V are totally umbilical in Mon. ii) If D is integrable and its leaves are totally geodesic in Mon then M' is D-geodesic. Conversely, if M'n is D-geodesic and tangent to the Lee field

of Mon then the holomorphic distribution of M' gives rise to a complex foliation of Mm whose leaves are totally geodesic in Mon.

Proof. Let D be integrable. Let S be a leaf of D and i : S

MI the

inclusion. We denote by OS and hs the Levi-Civita connection of i'g and the second fundamental form of i, respectively. Let us assume that hs = 0. By the Gauss formula

OxY=VXYED for any X, Y E D. Using this fact and the identities

h'=h+2g®Bl D°J=0 D = V -w®I+2g®B, where O denotes the symmetric product, we may conduct the following calculation for any X, Y E D and any Z E Dl

go(h(X, Y), JZ) = go(h'(X, Y), JZ)

- 2go(Bl, JZ)g(X, Y)

= go(DXY, JZ) - g(X,Y)wo(JZ)

= -go(JDzY, Z) - -g(X,Y)8(Z) _ -go (D°xJY, Z) - 2g(X,Y)O(Z)

12.5. DISTRIBUTIONS ON SUBMANIFOLDS

171

= -9(DxJY,Z)- 1g(X,Y)O(Z) = -9(V xJY, Z) - 29(X, JY)w(Z) - -g(X,Y)O(Z) hence

9o(h(X,Y),JZ)+ 2g(X,Y)O(Z) _ -2fl(X,Y)w(Z)

(12.49)

for any X, Y E D and any Z E Dl. The left-hand side of (12.49) is symmetric in X, Y while the right-hand side is skew symmetric. Thus both sides vanish, one leading to (12.48) and the other giving

1(X, Y)w(Z) = 0

for any X, Y E D and any Z E D. As p 0 0 it follows that w(D1) = 0, that is, B E D. Conversely, let us prove that (12.48) yields the involutivity of D. To this end, let X, Y E D and Z E D- L. As D° is torsion-free and almost complex we have

9([X,1'], Z) = 9o(J[X, Y], JZ)

= 9o (J { DX Y - D}°. X }, JZ) = go (DX JY - D°y JX, JZ) = go(h'(X, JY) - h'(Y, JX), JZ) because JZ E E(LY). At this point, substitution from h' = h + (1/2)g ® Bl furnishes

g([X, Y], Z) = go (h(X, JY) - h(Y, JX ), JZ) + SZ(X, Y)9(Z) 0

as a consequence of (12.48). Thus [X,YJ E V for any X,Y E D. The next step is to prove that under the assumptions (12.48) and alB = 0, where

it

: T(M') -, Dl is the natural projection, each leaf S of D is totally

geodesic in MI. Using D° = V° - wo O I + (1/2)go ® Bo and D°J = 0 we may conduct the following calculation for any X, Y E V and any Z E Dl:

9(hs(X, Y), Z) = 9(V xY, Z) = 9o(OXY, Z) = 9o(DXY, Z) - 1g(X,Y)9o(Bo, Z)

= 9o(DXJY, JZ) - 2g(X,Y)w(Z)

CHAPTER 12. SUBMANIFOLDS

172

9o(h'(X, JY), JZ) - 29(X,Y)w(Z) 9o(h(X, JY), JZ) + 2g(X, JY)9(Z)

29(X'Y)w(Z) - 29(X, Y)w(Z)

(for the last equality we used (12.48)). Thus, as Dl -+ S is the normal bundle of i : S -4 M' 1

hs = - 2 g ®c1B.

(12.50)

Thus (12.48) by itself yields the total umbilicity of i : S -' Mm. Finally, by B E D, our i : S -i Mm is also minimal. To prove the second part of statement i), let MI be a generic CR submanifold of lvfott. Then the normal bundle of Mm in Mo" is precisely J(D1) -+ M'. Consequently by (12.48)

h(X,Y) = -2g(X,Y)B1

(12.51)

for any X, Y E D. Since hs = 0, the second fundamental form of 41 o i : S -+ Mo" is precisely (12.51). Let us prove ii). To this end, suppose that V is integrable and its leaves are totally geodesic in Mo". Consequently,

VoYED for anyX,YED. LeteEE(T). Then go(h(X,Y),C) =go(V Y,e) =0 i.e. M' is D-geodesic. Conversely, suppose that Mm is D-geodesic. Then a calculation (similar to the proof of (12.50)) gives

t1[X,Y] = -12(X,Y)tB1 for any X, Y E D. Our Theorem 12.8 is completely proved.

12.6

Totally umbilical submanifolds

By a result of A. Bejancu, [18], there are no totally umbilical proper CR submanifolds of codimension > 1 of a Kahlerian manifold, except for the totally geodesic ones. As to the 1.c.K. case, we may state (cf. S. Dragomir, [76], p. 2):

12.6. TOTALLY UMBILICAL SUBMANIFOLDS

173

Theorem 12.9 Let Mm be a totally umbilical proper CR submanifold of the l.c.K. manifold Mpn. If dim(D1) > 1 and Mm is tangent to the Lee field of M02" then Mm is totally geodesic. Proof. We shall need the following:

Lemma 12.6 Let Mm be a CR submanifold of the l.c.K. manifold Mo". Then

AFXY = AFYX + 2 {wo(FY)X - wo(FX)Y}

(12.52)

for any X, Y E Dl D.

Proof. Let X, Y E Dl. As P(D1) = 0 and S2(Dl,Dl) = 0 the identity (12.9) may be written as

-P OXY = AFYX + t h(X, Y) + 2 {0(Y)X - g(X, Y)A}.

(12.53)

Let us interchange X and Y in (12.53) and subtract the resulting identity from (12.53). We obtain

-FIX, Y] = AFYX - AFXY + 2 {0(Y)X - 0(X )Y} (as V is torsion-free). Finally (12.52) follows from the involutivity of Di and because of 0(X) = wo(FX) for any X E Dl. The proof of Lemma 12.6 is complete.

Let H be the mean curvature vector of M'n in Mon. Then t H E Dl. Applying (12.52) for Y = tH gives

AFXt H = AFtHX + 2 {wo(FtH)X - wo(FX)tH}

(12.54)

for any X E Dl. Assume from now on that Mm is totally umbilical. Then ACX

= go(H, )X

for any C E E(LY). We may substitute in (12.54) to get

go(H, FX)tH = go(H, FtH)X + 2 {wo(FtH)X - wo(FX)tH}.

(12.55)

We wish to show that, under the given assumptions

tH=0.

(12.56)

CHAPTER 12. SUBMANIFOLDS

174

Assume that t H # 0 at some point x E Mm. By hypothesis q > 1. Thus we may choose X E VI so that g(tH, X) = 0 and X t 0. Taking the inner product of (12.55) with X leads to go(H, FtH) + 2wo(FtH) = 0

so that IItHII2 = Zwo(FtH).

(12.57)

Moreover, as Bl = 0, one has wo(E(fl) = 0. Thus (12.57) leads to 11t H11 = 0 at x, a contradiction. The identity (12.56) is completely proved.

In particular B' = 0 and (12.56) lead to

9o(H)=0. Next, let us apply (12.11) for £ = H. We obtain

-t OXH = A fHX - PAHX.

(12.58)

Let Y E T(M"'). Take the inner product of (12.58) with PY and use the fact that t is Dl-valued and the total umbilicity of MI in Mo". This procedure furnishes 0 = go(H, f H)9(X, PY) - II HII29(PX, PY) or, by observing that go(H, fH) = 0 II H II29(PX, PY) = 0

for any X,Y E T(Mm). By hypothesis p > 1. Consequently IIHII = 0 and Theorem 12.9 is completely proved. By Theorem 12.9, totally umbilical submanifolds (with h = 0) may not

occur among CR submanifolds (of a I.c.K. manifold) unless p = 0 or q E {0,1 } or Bl # 0. Before reporting in more detail on these cases, we wish to furnish a few examples. Let CH' be the complex Hopf manifold determined by A E R, 0 < A
2, then po = constant, cf. Lemma 9.1 in [296], p. 114. We may state the following (cf. S. Dragomir, [75], p. 200):

Theorem 12.13 Let M be a complex submanifold with a semi flat normal connection of the complex Hopf manifold CHa, dimt(M) = 2s. If n-s > 1 then M is a totally umbilical quasi-Einstein submanifold with a flat normal connection.

To clarify Theorem 12.13 we also state

Theorem 12.14 Any totally umbilical submanifold of a Hopf manifold is a quasi-Einstein manifold. To describe the structure of invariant submanifolds with a semi-flat normal connection further, we may state

Theorem 12.15 Let M be a complex submanifold with a semi-flat normal connection in CHa. i) If codim(M) = 2, s >_ 2, and (12.60) holds for some smooth function po : M -, (0, +oo), then M is a globally conformal Kahler manifold and its Lee form is given by w = -d log po. ii) If codim(M) > 2 and M is strongly non-Kahler then M cannot be an Einstein manifold.

Proof of Theorem 12.13. Let us combine (12.60) with the Ricci equation (12.15) and put rl = Jl; in the resulting equation. This procedure yields po 1(X,Y)IICII2 - g([AA,AjC]X,Y) = 0

(12.61)

12.6. TOTALLY UMBILICAL SUBMANIFOLDS

177

E E(W). At this point, by Lemma 11.2, we may substitute from (12.29) into (12.61) to yield

for any X, Y E T(M),

A +wo(f)A + 2 [poiieii2 + 2wo(e)2] I = 0.

(12.62)

Let us put n = s + p, p > 1, i.e. s < n, by hypothesis. Then either p = 1, i.e. M is a complex hypersurface of CH", or p > 2. For the last case let {V1, ... , Vp, JV1, ... , JVp} be a (local) orthonormal frame of E(W). Note that (12.62) is equivalent to

(' )2 = -1 PogII2I

(12.63)

(cf. (12.45)). Now, on the one hand,

A''A'' +A'A'. = 0 for i # j. On the other hand, by the R.icci equation (12.15) for

= Vi, 7 _

V,, we have Av; Av; - Av, Av, = 0 which yields

A'vzA'vj -AV

=0

such that A'vt A'vl =0

for i # j. Finally (by (12.63)) we obtain

A' = 0 or

A _ -2wo(W

(12.64)

that is M is totally umbilical and po = 0, i.e. Rl = 0. Since V°wo = 0 and Ilwo ll = 2 the Gauss equation (12.19) of M in CHI may be written

R(X, Y)Z = (X n Y)Z + +Ah(YZ)X - Ah(x,Z)Y +

4 {[w(X)Y - w(Y)X]w(Z)

+ [g(X, Z)w(Y) - g(Y, Z)w(X)]B

(12.65)

CHAPTER 12. SUBMANIFOLDS

178

for any X, Y, Z E T(M). Actually (12.65) holds for any submanifold M of CH' not necessarily a complex submanifold. Now substitution from (12.64) into the Gauss equation (12.65) and further contraction of indices lead to

Ric(X,Y) = [2(2s - 1) -

2

[[w[I2]g(X, Y)

S- lw(X)w(Y)

(12.66)

2

i.e. M is quasi-Einstein. Our Theorem 12.13 may be contrasted with a result of I. Ishihara, [143]. Proof of Theorem 12.15. We define a covariant derivative of R-L in the usual manner, cf. [296], p. 115. Then

E (VXR1)(Y, Z) = 0

(12.67)

XYZ

for any X, Y, Z E T(M). Here >xyZ denotes the cyclic sum over X, Y, Z. On the other hand (VxRl)(Y, Z) = X (po)Q(Y, Z)JZ; + po(OxR)(Y, Z)Jij.

(12.68)

Of course, in general, the induced connection of the complex submanifold M is not almost complex. In turn, we have

JOxY = VxJY +2{w(Y)JX - O(Y)X +O(X,Y)B+g(X,Y)A}.

(12.69)

Consequently, the Kahler 2-form 0 is not parallel, in general. Yet (12.69) yields

(V f) (Y, Z) = 1 [9(Z)g(X, Y) - 0(Y)g(X, Z) +w(Z)S2(X, Y) - w(Y)S2(X, Z).

(12.70)

Finally, part (i) of Theorem 12.15 follows from the more general

Proposition 12.3 Any complex submanifold, of a l.c.K. manifold, possessing (0, +oo) is globally cona semi-flat normal connection for some po : M formal Kdhler provided that M has complex dimension s > 2. Indeed, combining (12.68), (12.70) and (12.67) we obtain E X (po)11(Y, Z) = -3po(w n Sl)(X,Y, Z). XYZ

(12.71)

12.6. TOTALLY UMBILICAL SUBMANIFOLDS

179

Let us put Z = JY in (12.71). Since s > 2 one may choose X orthogonal to Y, JY. Thus dpo + pow = 0, Q.E.D. To prove the second part of Theorem 12.15 we suppose that M is Einstein, i.e. Ric = Ag for some A E R. By (12.66) we obtain 2IIwII2 _,\]X

[2(2s - 1) -

=

s 2

Iw(X)B

(12.72)

for any X E T(M). Let us put X = B in (12.72) and take the inner product with B. Since M is strongly non-Kahler (i.e. Sing(w) = 0) we obtain

2(2s-1)-

2s - 1 2

IIwII2=A.

Substitution in (12.72) now gives w(X)B = IIwII2X

which for X = JB furnishes IIwII = 0, a contradiction. We shall need the Yamabe functional of a compact submanifold M (dim(M) = 2s): E(,p),

I(cc) =

(12.73)

where E(w) = (aIId,II2 + IM

(JIW*1) 1/N

IRoIIN =

N =

1

2s

a

= 2(2s-1)

s-1

s-1

and p denotes the scalar curvature of M. Cf. [170]. The Yamabe invariant (a conformal invariant) of M is given by

po=inf{I(cp):cpEC°O(M),V2>0,(p0- 0} where COO(M) is the ring of all real-valued C°O functions on M. We may state (cf. S. Dragomir, [75], p. 201)

Theorem 12.16 Let M be a compact totally umbilical submanifold of real dimension 2s of the complex Hopf manifold CH, 'N. If s>1IIwI12

CHAPTER 12. SUBMANIFOLDS

180

and M has a nonpositive Yamabe invariant (µo < 0) then M is totally geodesic. Moreover, if, additionally, M is a complex submanifold then it is a generalized Hopf manifold, and it is a globally conformal Kahler manifold provided that the sectional curvatures of M are subject to

k(p) > 1 - 2{w(X)2+w(Y)2} for any p E G2(M) and any g-orthonormal basis {X, Y} in p. Here G2(M) M denotes the Grassmann bundle of all 2-planes on M. We shall prove Theorem 12.16 later on. We state a few additional results regarding the geometry of totally umbilical submanifolds of a complex Hopf manifold with further restrictions on their curvature.

Theorem 12.17 Let M2' be a real 2s-dimensional totally umbilical submanifold of CH, "\, 2 < s < n. If M2s is conformally flat then it has a vanishing scalar curvature. By a theorem of S.I. Goldberg & M. Okumura, [114], if M2s is conformally flat and has constant scalar curvature p and the length of the Ricci tensor is < p(2s -1)-1/2 , s > 2, then M2s is a space-form. If in turn M2s is a totally umbilical submanifold of CHa then we have liRicJi = 0 if and only if M2s is a surface, i.e. s = 1. We actually prove

Theorem 12.18 Let M2,' s > 2, be a conformally flat totally umbilical submanifold of CH,\. Then M2s is never a space of constant curvature.

Theorem 12.19 The only totally umbilical complex submanifolds of zero scalar curvature in a complex Hopf manifold are the totally geodesic flat surfaces.

Theorem 12.20 In a complex Hopf manifold there do not exist any projectively flat totally umbilical submanifolds with zero scalar curvature and nowhere vanishing 1-form w. In [76], p. 182, one has obtained Theorem 12.21 A complex Hopf manifold CH, contains no totally geodesic elliptic space form MI (k) , k > 0, m > 2, such that w2 # 0, for any x E Mm(k).

-

12.6. TOTALLY UMBILICAL SUBMANIFOLDS

181

Note that Theorem 12.21 is not a corollary of Theorem 12.18 due to the restriction on curvature imposed there.

Let M2s be a submanifold of CH',\. By (12.65) we may compute the Ricci, and therefore the scalar curvature of M2s: 2(n-s)

p = (2s - 1)(2s -

2IIwII2)

+

[trace(Ab)2

- trace(Ab)]

.

(12.74)

6=1

If M2s is totally umbilical then S-21

Ric

+ (2s

U) Ow

- 1 - 1 IIWI12 + (2s - 1)IIH112) g

(12.75)

and Theorem 12.14 is proved. Also (12.74) reduces to

p = (2s -1)(2s -

21

2 11W112)

+ 2s(2s -1)IIH112.

(12.76)

Furthermore, we prove Theorem 12.16. If po _< 0 then there exists W E C°O(M2s), cp > 0, cp # 0, such that I(cp) < 0. Using (12.73) and (12.76) we obtain *1 S-1 Ji,12.'

+2s fM23 1p211HII2 * 1 < 0.

Since s > IIWII2/4 we obtain dcp = 0, i.e. cp = const. We always assume M2s to be connected. Yet cp 0- 0 yields IIHII = 0. Thus h = 0 since M2s is totally umbilical. Moreover, we obtain that w is parallel. Consequently, if M2s is a complex submanifold, then it is a l.c.K. manifold with a parallel Lee form, i.e. Vaisman manifold. To prove the last statement in Theorem 12.16, let K be the curvature of D (the connection induced on M21 by the Weyl connection of the ambient complex Hopf manifold). Then g(K(X, Y)Z, U) = g(R(X, Y)Z, U)

-2{L(X, Z)g(Y, U) - L(X, U)g(Y, Z) -L(Y, Z)g(X, U) + L(Y, U)g(X, Z)} +wo(h(X, Z))g(Y, U) - wo(h(Y, Z))g(X, U)

CHAPTER 12. SUBMANIFOLDS

182

+wo(h(Y, U))g(X, Z) - wo(h(X, U))g(Y, Z) -4(IIwoII2 + IIB1II2)(g(Y, Z)g(X, U) - g(X, Z)g(Y, U))

(12.77)

for any X, Y, Z, U E T(M2s). Let (x') be real analytic local coordinates on CHI,\. We recall that the Boothby metric go is locally conformal to the (local) Kahler metrics go = bldx'dx', i.e. go = IxI-ego. Let f be the restriction to M2s of 2 log IxI. Also let k' be the sectional curvature of the metrics W'go where :M2s -+ CH' is the given immersion. By (12.77) we obtain

exp(-f)k'(p) = k(p) - 1 + 2 {w(X)2 + w(Y)2}

(12.78)

for any p E G2 (M2.) and any g-orthonormal basis {X, Y} of p. The condition

in Theorem 12.16 yields k' > 0. As po < 0 we may apply a result of I. Vaisman (i.e. Theorem 3 in [2701, p. 281) to conclude that M2s is a g.c.K. manifold. Next, we look at conformally flat submanifolds of a complex Hopf manifold.

Let M2s be a totally umbilical submanifold of CH, \, s > 2. We

consider the Weyl conformal curvature tensor of M2s:

W(X,Y) = R(X, Y)

- 2(31

1)

(QX AY+X AQY)

P

+4(s

- 1)(2s - 1)X AY

(12.79)

for all X,Y E T(M2 ). Here Q = (Ric)e or by (12.75), QX = [(2s - 1)(1 + IIHII2)

-s

2

- 1 IIwII2)X

Iw(X)B.

(12.80)

Taking into account (12.76) and (12.80), the expression of the conformal curvature tensor (12.79) becomes W(X) Y) = R(X, Y) + 2(s1 1) [iIwiI2 - (3s - 2)(1 + IIHII2) X A Y +4 [w(Y)X A B - w(X)Y A B1.

Finally, one may use the Gauss equation (12.65) and Ah(y,.)X - Ah(X,.)Y = IIHII2X A Y.

(12.81)

12.6. TOTALLY UMBILICAL SUBMANIFOLDS

183

Consequently (12.81) reduces to

W(X,Y) =

2(31

1) [iiwi2 - s(1 + IIHII2)] X A Y

(12.82)

for any X, Y E T (M2. ). Now W = 0 if and only if IIwII2 = 4s(1 + IIHII2)

(12.83)

i.e. if and only if p = 0, cf. our (12.76). This proves Theorem 12.17. As

remarked before, Theorem 2 in [114), p. 234, may not be applied to our case since (12.75) yields

1

+IIHII2 =

(s - 1)[2s(2s -1)1112.

(12.84)

To prove Theorem 12.18, by (12.65) we obtain the expression of the RiemannChristoffel tensor of the (totally umbilical) submanifold M2a, i.e.

g(R(X,Y)Z,W) = (1 + IIHII2)(g(Y, Z)g(X,W) - g(X,Z)g(Y,W)) +4 {[w(X)g(Y, W) - w(Y)g(X, W)]w(Z)

+[g(x, Z)w(Y) - g(Y, Z)w(X))w(W)}, which yields the sectional curvature of M2a:

k(p) = 1 + IIHII2 - 4{w(X)2+w(Y)2} for any p E G2 (M2,) (spanned by the orthonormal vectors X, Y). Suppose now that M2a is a space-form, i.e. k(p) = c, c E R, for all p E G2(M23). Then p = 2s(2s - 1)c.

Thus W = 0 yields c = 0 or 4(1 + IIHII2) = w(X)2 + ,(Y)2.

Let Y = IIBII-1B. Then s = 0 by (12.83), a contradiction. Let us now look at totally umbilical submanifolds of zero scalar curvature.

Let M2a be a complex submanifold of CHa with p = 0 and h = g 0 H. By

CHAPTER 12. SUBMANIFOLDS

184

(12.27) the mean curvature vector H of a complex submanifold of a 1.c.K. manifold is given by

H = -1B1. 2 Also IIwoII2 = IIWI12 + IIB1112 and IIwo11= 2 yield (by (12.83))11w112 = 8s/(s+

1). But 4 - 8s/(s + 1) _> 0 yields s = 1 and H = 0, such that M2 is a minimal surface in CH' Thus h = 0 and (by Theorem 12.22) M2 is flat. Thus Theorem 12.19 is completely proved. To establish Theorem 12.20 let

P(X, Y) = R(X, Y) -

s

1 1(X A Y) o Q

(12.85)

be the projective curvature tensor of M2,, s > 2. By (12.80) and (12.65) we put (12.85) in the following form

P(X,Y)Z =

s

1 1

[1 IIWI12 -s(1 + IIH112)1 (X A Y)Z

+ 1 f [w(Y)X - w(X)Y]w(Z)

+[g(X, Z)w(Y) - g(Y, Z)w(X)]B}.

(12.86)

If p = 0 then by (12.83) and P = 0 we obtain (w(Y)X - w(X)Y)w(Z) = (g(Y, Z)w(X) - g(X, Z)w(Y))B.

(12.87)

We put Z = B in (12.87).

0 everywhere gives w(Y)X Then IIw11 w(X)Y = 0 which together with (12.87) gives g(Y,X) = WI12w(I')w(X) and in particular w(Y) = IIw1111Y11 This yields IIYIIX - IIXIIY = 0 for any

X, Y, a contradiction.

Theorem 12.22 Let M', m > 2, be a totally geodesic submanifold of the (non-Kdhler) g.H.o manifold Mo (with [[wolf = 2c, c E R). Then i) Mm has a flat normal connection (R1 = 0). ii) The induced 1-form w is parallel. Thus either w = 0 and then MI is an elliptic space form, or w 0 everywhere and then M' is tangent to the Lee field (B1 = 0) of M'. iii) If in = 2 then M2 is a real space form M2(k) where either k = c2 if M2(k) is normal to the Lee field of Mon, or k = 0 if M2(k) is tangent to the Lee field.

12.6. TOTALLY UMBILICAL SUBMANIFOLDS

185

Proof. Cf. [77], p. 182, and [86], p. 3-4. The fact that Rl = 0 was demonstrated at the end of Section 11.1 (and holds for totally geodesic submanifolds in any l.c.K.o manifold - not necessarily with a parallel Lee form). The parallelism of w follows from (12.39). Then either w = 0 or w # 0 everywhere on Mm.

Let us go back for a moment to (12.19) (true for any submanifold Mm in a 1.c.K.0 manifold Man). If MOn is a g.H.o manifold (i.e. it additionally satisfies Vowo = 0) then (12.19) becomes

R(X,Y)Z = c2(X AY)Z +4 {w(Z)[w(X)Y - w(Y)X] +[g(X, Z)w(Y) - g(Y, Z)w(X)]B} +Ah(y,Z)X - Ah(X,Z)Y

(12.88)

where IIwoI] = 2c. If additionally h = 0 then (by (12.39)) the Gauss equation (12.88) becomes

R(X,Y)Z = c2(X AY)Z

+4{[w(X)Y - w(Y)X]w(Z) +[g(X, Z)w(Y) - g(Y, Z)w(X)]B}.

(12.89)

Contraction of indices in (12.89) then gives Ri c = [( m - 1) c2 - 1 II w II2] g

- m4

2 w ® w.

(12 . 90)

If w = 0 then (12.89) furnishes

R(X,Y)Z = c2(X AY)Z

i.e. M' is an elliptic space form M'(c2). If w 54 0 then Sing(w) = 0 and we need to show that B'L = 0.

To this end we look at the Codazzi equation (12.20) of a submanifold Mm in a 1.c.K.0 manifold. As h = 0 and B is parallel, we have

v1Bl = 0 so that (12.20) becomes

0=

4{g(X,Z)w(Y) - g(Y, Z)w(X)}B'.

(12.91)

186

CHAPTER 12. SUBMANIFOLDS

Assume that Bl # 0 at some point xo E Mm. Then we may take (as m > 2) Z = X # 0, g(X,Y) = 0. Then (12.91) yields IIXII2w(Y) = 0 for arbitrary

Y, so that w = 0 at xo, a contradiction. Thus B' = 0. So ii) in Theorem 12.22 is proved. To prove iii), we only need to check that whenever Bl = 0 the real surface M2 is flat. Indeed, if Bl = 0 then IIwII = IIwoII = 2c and (12.90) becomes

Ric = (m-2) (c2g -

4WOW )

.

(12.92)

Thus m = 2 yields Ric = 0 and, since M2 is a surface, R = 0, Q.E.D.

Finally, we ought to prove Theorem 12.21. By ii) in Theorem 12.22 we only need to check that when Sing(w) = 0 the submanifold cannot be an elliptic space form. We have already seen that this is the case for real surfaces. For arbitrary dimension m > 2 we shall show that the sectional curvature may vanish for certain tangent 2-planes. First, by (12.65) we have k(p) = 1 - 4 [w(X)2 + w(Y)2] +go(h(X, X), h(Y, Y))

- IIh(X,Y)II2

(12.93)

for any p E G2(MI) and any orthonormal basis {X,Y} of p. Let h = 0. Let po E G2(Mm) spanned by X, 1IXII = 1, and Y = B/2, g(X,B) = 0. Since (by Theorem 12.21) Bl = 0 one has IIYII = 1. By (12.93) we obtain k(po) = 0 so that Theorem 12.21 follows from the more general statement

Proposition 12.4 In a complex Hopf manifold CHa there are no totally geodesic submanifolds, of positive sectional curvature, and so that Sing(w) _ 0.

Chapter 13

Extrinsic spheres 13.1

Curvature-invariant submanifolds

An extrinsic sphere of a R.iemannian manifold M is a totally umbilical (h = g ® H) submanifold M of M so that its mean curvature vector is nonzero

(H # 0) and parallel in the normal bundle (O'H = 0). For instance, any circle of M (in the sense of K. Nomizu & K. Yano, [204], p. 163) is either a geodesic or a 1-dimensional extrinsic sphere of M. Also, if M is an extrinsic

sphere in M then every circle in M is a circle in k and conversely, cf. Theorem 2 in [204], p. 166. Extrinsic spheres in Kahlerian manifolds have been widely studied, cf. e.g. B.Y. Chen, [54]-[56]. We report on the knowledge on extrinsic spheres in l.c.K. manifolds. To start with, we state (cf. [49])

Theorem 13.1 Let M2i-1 be an orientable totally umbilical real hypersurface of the complex Hopf manifold CHa , n > 2. Then M2i-1 has a parallel mean curvature vector if and only if for each x E Men-1 either wx = 0 or Men-1 is tangent at x to the Lee field of CH, \. We shall prove Theorem 13.1 later on. Let M be a submanifold of the Riemannian manifold M. Let R be the curvature 2-form of M. Then M is a curvature-invariant submanifold (cf, the terminology of K. Ogiue, [207], p. 389) if for any x E M and any u,v E TT(M) the tangent space Tx(M) is invariant under the curvature transformation Rx(u, v) : Tx(M) -+ T. (M),

that is Rr(u, v)TT(M) 9 Tx(M).

CHAPTER 13. EXTRINSIC SPHERES

188

Also M is called strongly curvature-invariant if TZ(M) is invariant under Rx(u, v) for any u, v E T, (k) , x E M. For instance, any submanifold of a real space form is curvature-invariant, cf. 12071, p. 390. Also, each extrinsic sphere is curvature-invariant, by the Codazzi equation, see e.g. B.Y. Chen, [54], p. 329. We may finally mention that since any curvature-invariant submanifold of a complex space form is either holomorphic or totally real, cf. B.Y. Chen & K. Ogiue, [59], p. 260, the totally umbilical submanifolds of complex space forms have been completely classified, see Theorem 1 in [60], p. 225. As to (strongly) curvature-invariant submanifolds of complex Hopf manifolds (with the Boothby metric) we may state

Theorem 13.2 Let M'" be a real m-dimensional (where 1 < m < 2n), strongly curvature-invariant submanifold of CHa , n > 1. Then either Mm is normal to the Lee field of CHa or Mm is a real hypersurface of CHa .

Theorem 13.3 Let Mm be a totally umbilical submanifold of the complex Hopf manifold CH, \. If Mm is tangent to the Lee field of CHa then MI is either totally geodesic or an extrinsic sphere.

Let us prove Theorems 13.1 up to 13.3. Let M2"-1 be an orientable real hypersurface of CHa . Let N be a smooth field of unit normal vectors on

M2r-1. Then e _ -JN is tangent to M2s-1. Also we shall use the 1-form z on M2r-1 given by 17(X) = go(FX,N)

for any X E T (M2i-1) (so that 719 = l;). Actually (P, l;, r7, g) is an almost contact metric structure on Men-1, Cf. Y. Tashiro, [253]. Set A = AN

for simplicity. The Gauss and Weingarten equations of the given immersion

T : Men-1 -. CHa read

for any X,Y E

O°tY = OxY+g(AX,Y)N

(13.1)

Vo N = -AX

(13.2)

T(M2n-1). By (12.9) in Chapter 12

one obtains

(VxP)Y = g(Y)AX -g(AX,Y)C-

-2{w(Y)PX - w(PY)X + 12(X,Y)B - g(X,Y)PB+

13.1. CURVATURE-INVARIANT SUBMANIFOLDS

+wo(N)(g(X,Y)£ - 77(Y)X])

189

(13.3)

for any X,Y E T(M2n-1). As h(X,Y) = g(AX,Y)N the Gauss equation of IF may be written R(X, Y)Z = (X A Y)Z + g(AY, Z)AX - g(AX, Z)AY

+4{(w(X)Y - w(Y)X]w(Z) + [g(X, Z)w(Y) - g(Y, Z)w(X)]B}

(13.4)

for any X,Y,Z E T(M2n-1). Also, using Bl = wo(N)N, the Codazzi equation of %F may be written

(VxA)Y - (VyA)X = 4{w(Y)X -w(X)Y}wo(N)

(13.5)

If M2n-1 is totally umbilical then

H=

1

2n - 1

vN

,

AX = go(H, N)X

where v = trace(A). The last equation may be also written as A

2nv

II.

(13.6)

Clearly (13.6) yields

(VxA)Y = 2n1 1X(v)Y

(13.7)

for any X,Y E T(M2n-1). Using (13.7) and the Codazzi equation (13.5) we obtain 1 {2n1

1X(v) + w(X)wo(N) }Y

2n1 1Y(v) + 4w(Y)wo(N)} X = 0.

(13.8)

As n > 2 one may choose J(YJ' = 1, g(X,Y) = 0. Taking the inner product of (13.8) with Y one obtains

dv = -

2n - 1 4

wo(N)w.

(13.9)

At this point one may prove Theorem 13.1. Since VlN = 0 and v = (2n - 1)go(H, N) we have dv = 0 if and only if H is parallel in the normal

CHAPTER 13. EXTRINSIC SPHERES

190

bundle. On the other hand (by (13.9)) (dv),, = 0 if and only if either wx = 0 or Bo,.,, is tangent to M2i-1, Q.E.D. We mention that identities (13.1)-(13.9) will be used for an in depth study (to be performed in Chapter 14) of the geometry of real hypersurfaces in a complex Hopf manifold, such as types, principal curvatures, homogeneous real hypersufaces, etc. Let M'" be a submanifold of CH',k. The Codazzi equation of the given immersion T : M'" CHa may be written as follows:

nor{Ro(X, Y) Z) = 4 {g(X, Z)w(Y) - g(Y, Z)w(X)}Bl.

(13.10)

Therefore, any submanifold of CHa which is tangent to the Lee field is curvature-invariant. Let Fo be the canonical foliation of CHa (i.e. T(Fo) = Ker(wo)). By (13.10) any leaf of Fo is a curvature-invariant submanifold of CHa . Conversely, one has

Proposition 13.1 Let M', m > 2, be a curvature-invariant submanifold of CH. \. Then either M"` is tangent to the Lee field of CHa or, if Bi # 0 at some x E Mm then x E Sing(es).

Proof. If Mm is curvature-invariant then (by (13.10)) one has {(u, w)ws(v) - (v, w)wx(u)}Bz = 0

(13.11)

for any u, v, w E T. ,(M') and X E Mm. Here (,) = gy. We distinguish two cases. Either Bl = 0 or there exists xo E MI so that B .,L. 34 0. If this is the case, let u E Txo(Mm) be arbitrary. Since m > 2 one may consider v = w, jlvii = 1, (u,v) = 0, v E ,,O(M). By (13.11) one obtains wx0 (u) = 0 for all u, Q.E.D. Let us prove Theorem 13.2. Suppose Mm is strongly curvature-invariant. Then for any tangent vector fields k, Y on CH, \, respectively for any tangent

vector field Z on M', one has {go(Y, Z)

- 4wo(Y)w(Z)}nor(ff)

-{go (.k, Z) - 1Wo(X)w(Z)}nor(Y)

+4 {go(X, Z)wo(}') - go(Y, Z)wo(fX)}Bl = 0

(13.12)

as a consequence of (12.65). K. Ogiue, [207], p. 393, has shown that there exist no strongly curvature-invariant submanifolds in a Riemannian manifold

13.1. CURVATURE-INVARIANT SUBMANIFOLDS

191

of nonzero constant curvature. Following the scheme of the proof of Theorem

3.4 in [2071, p. 393, let k = X = Z, Y = , where X is tangential while is normal. Then (13.12) leads to {4w(X)2

- IIXII2} + 4IIXII2wo(C)Bl = 0.

(13.13)

We distinguish two possibilities. Either w - 0, i.e. Mm is normal to Bo, or there exists xO E Mm so that w,,o # 0. If this is the case then BO 34 0 and, since m > 2, we may choose u E T2p(Mm) , lull = 1, (u, Bso) = 0. Apply (13.13) for X10 = u. It follows that wo,xo(z)B u = 4z

(13.14)

for any z E E(T)xo. Clearly Bo # 0, otherwise, by (13.14) one would have m = 2n, a contradiction. Thus, by (13.14), E('I)xo is spanned by B .,-,L,, i.e. codim(Mm) = 1, Q.E.D. Let Mm be a totally umbilical submanifold of CH, "\. By the Codazzi equation and total umbilicity,

nor{Ro(X, Y)Z} = g(Y, Z)V H - g(X, Z)VYH.

(13.15)

As m _> 2, for fixed X let us choose Y such that 11Y11 = 1, g(X, Y) = 0. Then (13.15) yields nor{Ro(X, Y)Y} = VkH. This and (13.10) yield

VXH = -4w(X)Bl.

(13.16)

The situations in Theorem 13.3 now correspond to H = 0 and H

0,

respectively.

Let M23 be an even-dimensional extrinsic sphere of the l.c.K. manifold Mo', having a flat normal connection. If wo - 0 (i.e. the ambient space is Ki hlerian) then the simply connected extrinsic spheres with a flat normal connection have been classified by B.Y. Chen, [54). Actually, any such M2a

is isometric to the sphere S2a(l/k) of radius 1/k, where k = IIHII = const. See Theorem 1 in [54], p. 328. The classification problem for the case of a l.c.K. ambient space has only been partially solved (cf. S. Iamq & K. Matsumoto & L. Ornea, [134]) as yet. The key ingredient in the proof of Theorem 1 of [541, p. 328, is a result of M. Obata, [2051, p. 334, asserting that a complete Riemannian manifold Mm of dimension m > 2 is isometric to the sphere S'^(11r) if and only if the differential equation

Vdf = -r2g f

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192

admits some nonconstant solution f E C°O(M). Let M2' be a simply connected extrinsic sphere in the l.c.K. manifold Mo", having a flat normal connection. Before reporting on S. Ianu§ & K. Matsumoto & L. Ornea's classification, let us observe that the differential equation

V df + 2 (df) ®w = -k2g f

(13.17)

admits nonconstant solutions f E C°O(M2s) provided that M2' is tangent to both the Lee and anti-Lee vector fields of Mo", cf. S. Dragomir, [771, p. 387-388. One expects that the existence of nonconstant solutions of (13.17) may be used to show that M2' S2., (11k) (an isometry). While

this is not as yet available, let us prove the result stated in the remark above. Since M2s is taken to be simply connected, the assumption Rl = 0 is equivalent to the existence of a frame in the normal bundle consisting of mutually orthogonal parallel unit vector fields. Let N = k-1H. Then N is a parallel unit normal field. Choose a frame {Na}1 3 we have rri(M)

rrj(P(N))

If 7r1(P(N)) = 0 and dimc P(N) > 2 then either ir1(M) = 7r2(M) = 0 or 7r1 (M) = 7r2 (M) = Z. Thus, in general, M = M(F1i , F,) is not even homeomorphic to a standard sphere.

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204

As already mentioned, by a result of A. Bejancu, [18], in a Kahlerian ambient space totally umbilical CR submanifolds only occur among totally real submanifolds (p = 0) or in CR codimension 1, e.g. among real hypersurfaces of Man. However, by a result of Y. Taschiro & S. Tachibana, [254], neither elliptic nor hyperbolic complex space forms possess totally umbilical real hypersurfaces. Cf. also [296], p. 154. The situation, as demonstrated by Theorem 12.9, is sensibly different when wo # 0. Let Mon be a g.H. manifold (ilwo11 = 2c, c E (0,+oo)) and (Mm,D) a totally umbilical CR submanifold of Mpn. Let X, W E Dl. Then (12.8) yields

(V J)W = 2{wo(JW)X -w(W)JX +g(X,W)JBo}.

(13.41)

The left-hand side of (13.41) may be computed by using the Gauss and Weingarten formulae. Then (13.41) becomes

-go(H, JW)X + V*JW - JVxW - g(X, W)JH = 2{wo(JW)X - w(W)JX +g(X,W)JBo}. Take the inner product with X. This gives go(H,JW)IIX112 = g(X, W)go(H, JX)

-

2{wo(JX)jlXIj2 -g(X,W)wo(JX)} (13.42)

for any X, W E D1. Define E E E(T) by setting

E=H+1B and let Sing(E) = {x E Mm : EE = 0}. Let ,u (IF) C E(`I')

be the orthogonal complement of JDl in E('). Then Jxp('I')x = p('I')x for any x E Mm. Define

N={xEM':E.Ep(W)x}. We distinguish two cases, as I) Mm - N # 0, or II) Mm = N. If M' - N # 0 then let x E Mm - N. It follows that Ex 0 and E2 is not orthogonal to

J.,Di. That is (Er,Jxw)

0for some WEDz,w#0. Here

Kx be the orthogonal complement of Rw in P.1%

g..'. Le

13.3. COMPLETE INTERSECTIONS

205

Lemma 13.7 Let (Mm, D) be a totally umbilical CR submanifold of the 1.c.K. manifold Mpn. If M' - N 0 then dimR(D') = 1.

Proof. The proof is by contradiction. Assume that K= # {0}. Let then v E Kx , v # 0. Clearly (v, w) = 0. Then (13.42) yields (H., Jxw) _ -2wo(JJw)

that is (E_, Jxw) = 0,

a contradiction. Thus D,, = Rw. Let us look now at the case M'n = N. We distinguish two subcases, as

11.1) M' = Sing(E), or II.2) M' - Sing(E) # 0. Case II.1 leads to ii) of Theorem 13.8. If Mm - Sing(E) # 0 then E # 0 and E E µ(W). Thus

JE E ju (T),aswell. Lemma 13.8 Let (Mm, D) be a totally umbilical CR submanifold of the l.c.K. manifold Mon. If MI = N and M' - Sing(E) # 0 then OXE E z('P)

(13.43)

for any X E T(M'n).

Proof. Let Y E Dl. Using Vogo = 0, the identity (13.41), and the Gauss and Weingarten formulae, we may perform the following calculation

go(V E, JY) = X(go(E, JY)) - go(E, V9 JY)

_ -go(E, JOXY + 2 {wo(JY)X - w(Y)JX + g(X, Y)JBo}) = g(X,Y){go(JE, H) - Zgo(E, JBo)} = 0 and Lemma 13.8 is proved. Next, identity (12.8) for X E V and Y = E becomes

(V CJ)E = 2 {wo(JE)X - wo(E)JX}.

(13.44)

Let us apply tan to both sides of (13.44) and use Lemma 13.8. We obtain

{go(JE, H) + 1wo(JE)}X = {go(E, H) + 2wo(E)}JX,

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206

that is, 0 = IIEII2JX and consequently D = {0}, i.e. Mm is anti-invariant. Let us look again at case I, i.e. E # 0, E E µ('Y), and q = 1. Assume

that m > 5. Then p > 2. Let X E V be arbitrary. Due to our assumption on dimensions, we have Y E V, Y # 0, so that g(X, Y) = g(X, JY) = 0. By the Codazzi equation

go(Ro(X,Y)Z,f) =g(Y,Z)go(V4H,.) -g(X,Z)go(Vj H,f)

(13.45)

we obtain

go(Ro(JX,Y)JY,e) = 0

(13.46)

for any E E(W). From now on we confine ourselves to the case of a g.H. manifold M02" (as an ambient space). Let D° be the Weyl connection of Mon. Let Ko be the curvature tensor field of D°. Then K° is related to Ro by (12.13). Using D°J = 0 and (12.13) we may derive the identity Ro(JX, JY)Z = Ro(X, Y)Z +4 {[wo(JX)JY - wo(JY)JXJwo(Z) +Igo(JX, Z)wo(JY) - go(JY; Z)wo(JX)JBo}

-c2{go(JY, Z)JX - go(JX, Z)JY} -4{Iwo(X)Y - wo(Y)XJwo(Z) +Igo(X, Z)wo(Y) - go(Y, Z)wo(X )JB0}

-c2{go(Y, Z)X -go (X, Z)Y}

(13.47)

for any X, Y, Z E T(M). Let X E V be arbitrary, and Y E V such that Y # 0, g(X, Y) = g(X, JY) = 0 (as in (13.46)). Then (13.47) yields

Ro(JY,X)JY = Ro(JX,Y)JY

+1 [w(JY)X +w(Y)JX - w(JX)Y - w(X)JYJw(JX) -Ic2X

- 1w(X)BoJIIYII2.

Let us take the inner product with e. We obtain by (13.46) go(Ro(JY,X)JY,e) =

(13.48)

13.3. COMPLETE INTERSECTIONS

207

At this point, we use again the Codazzi equation (13.45) to compute the left hand side of (13.48). This procedure yields -IIYII29o(V

4IIYII2w(X)wo(C)

or

VXH = -4w(X)B-L

(13.49)

for any X E D. As V°wo = 0 we may use the Gauss and Weingarten equations to derive the identities

OXB = go(H, B1)X

(13.50)

V B1 = -w(X)H

(13.51)

for any X E T(Mm). To end the proof of iii) in Theorem 13.8, let X E Dl and set C = JX E E(T ). Let

E µ(W) and Y E D, Y # 0. Then, again by the Codazzi equation

(13.45)

go(Ro(X, Y)JY,

0

(13.52)

for any XED--,YEVand CEµ(W). Using D°J=0and(12.13)wemay derive

Ro(X,Y)JZ = JRo(X,Y)Z +4 {[wo(X)Y - wo(Y)XJwo(JZ) +[go(X, JZ)wo(Y) - 9o(Y, JZ)wo(X )JBo}

+c2{go(Y, JZ)X - go(X, JZ)Y}

-4 {[wo(X)JY - wo(Y)JXJwo(Z) +[go(X, Z)wo(Y)

- go(Y, Z)wo(X)JJBo}

-c2 {go(Y, Z)JX - go(X, Z)JY}

(13.53)

for any X, Y, Z E T(Mon). Then, for any X E Dl , Y E V we obtain by (13.53)

R0(X,Y)JY = JRo(X,Y)Y + [w (X )Y - w(Y)X Jw(JY) 4

CHAPTER 13. EXTRINSIC SPHERES

208

I ((w(X)JY - w(Y)JX]w(Y) - IIYII2w(X)JBo} -c2 IIYII2JX.

(13.54)

Let us take the inner product with Je and use (13.52) to obtain -4IIYII2w(X)wo(()

(13.55)

for any X E D- L, Y E D and any £ E Finally, let us use the Codazzi equation (13.45) to compute the left hand side (13.55). Then (13.55) becomes

-4w(X)woV)

that is

V H + 4w(X)B1 E JD1

(13.56)

for any X E D1. Next, let us take the inner product of (13.54) with JX to obtain (since go(Ro(X, Y)JY, ) = 0 by the Codazzi equation (13.45))

go(Ro(X,Y)Y,X) =

c2IIXII2IIYII2

-4 {IIXII2w(Y)2 + IIYII2_(X)2}

(13.57)

for any X E D1, Y E D. Let us replace Y by Y + JY in (13.57). This procedure furnishes

go(Ro(X,Y)JY,X) _ -4IIXII2w(Y) +w(JY)

(13.58)

for any X E D- L, Y E D. Next, take the inner product of (13.54) with X and use (13.58) to derive go(Ro(X,Y)Y,C') = -4IIYII2w(X) +wo(C')

(13.59)

for any X E D1, Y E D. By the Codazzi equation (13.45) one may compute the left hand side of (13.59). Thus (13.59) becomes

9o(VkH,C') = -4w(X)woW) so that as q = 1

V H+ 1w(X)B1

E p(W)

(13.60)

13.3. COMPLETE INTERSECTIONS

209

for any X E Dl. Then (13.56), (13.60), and JD' fl µ(') = {0} lead to

V H = -1w(X)Bl

(13.61)

for any X E D-L. Finally (13.49) and (13.61) lead to

V-LH = 4w ® Bl.

(13.62)

If Mm is normal to the Lee field (B = 0) then (13.62) shows that Mm is an extrinsic sphere in Mo". Compare with [73] for the case of a Kahlerian ambient space. There, use is made of the classification of extrinsic spheres in a Kahlerian manifold performed in [54] and [292]. As to the l.c.K. case, statement iii) in Theorem 13.8 follows from Theorem 13.4. We summarize our discussion in the following table:

Mm=N Mm = Sing(E)

Mm - Sing(E) # 0

B1=0==> h=0

D={0}

Mm-N#0 dimR, Dl = 1

m>5=* H, 01

O1H=--nw®B.L

0

Mm

Sm(1/k)

Let us now discuss a class of examples of extrinsic spheres which, in general, is are not even homeomorphic to standard spheres. Note that normal to the Lee field B of CHa locally given by B = zj8/8zj +-zi8/r9-z1. With the notations above, let N = N(F1, , Fr) be given by %F(S2m-1)

N=Nns2m-1

Then, by a result of B.Y. Chen, [56], N is an extrinsic sphere in N. Note that f : CH, -r S1 X S2m-1 descends to a C°O diffeomorphism:

,GI(Fl,...

Fr,)

S' xN(Fi,...SFr).

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210

Clearly k is a l.c.K. manifold (with the structure inherited from CHa as a complex submanifold). Since

zj a = it follows that M is tangent to the Lee field of CHI. At this point we may prove that M = M(Fi, . , F,) is an extrinsic sphere in k = M(Fi, - - -, F,). It suffices to prove this locally. That is, we wish to show that N is an extrinsic sphere in N, where N is endowed with the metric induced from

(C' - {0}, IzI-2bij). Let U be the restriction of log IzI-2 to N. Let G = exp(U)G where G is induced on N from (C' - {0}, bi,). Let j : N N be the inclusion. Set u = U o j. Total umbilicity is invariant under conformal transformation of the metric on the ambient space. Thus N is totally umbilical in (N, G). To check that the mean curvature vector of N in (N, G) is parallel in the normal bundle we recall (cf. (12.44)) that

H = exp(u){H+ I 2I

BI-}

where H, k are respectively the mean curvature vectors of N in (N, G) and (N, G). Also Bl = nor(Bo) and Bo = tan(B). The Lee form wo = dlog IzI-1 of (C' - {0}, IzJ-2bij) is parallel. As N is tangent to B it follows that N is a g.H. manifold. Thus (13.50)-(13.51) hold on N and

DXH=0 may be written

V*H=-4w(X)H=0 because N is normal to Bo. Here V1, Dl are respectively the normal connections of N in (N, G) and (N, G). To show that in general M = M(F1, - - -, F,) is not even homeomorphic to a sphere, we may compute its homotopy groups following (561, p. 204-205. To this end, let D be the lattice given by D = {27ria + (log A)b : a, b E Z}

(where log A E {log Ian + i(2kir + arg(A)) : k E Z}) and set

Tc = C/D. Then T1 acts on CHa by (w + D, 7r(z)) r-' 7r(eWz)

13.3. COMPLETE INTERSECTIONS

211

and CH' becomes a Tc-principal bundle over CPsi-1. There is a naturally induced action of Tc on M. In particular S1 acts on M and M becomes a S'-principal bundle over P(N). Now, applying the homotopy sequence of P(N) we obtain the following a fibration to the circle bundle S1 -+ M exact sequence: 7rj(S')

-

7rj(M) - 7rj(P(N)) - 7rj-1(Sl)

7r2(S1)

...

ir2(M) - 7r2(P(N))

(13.63) - 7r1(S1) -' 7r1(M) -'7r1(P(N)). It is well known that the j-th homotopy group 1rj(S1) vanishes for j > 2 and 7r1(S1) = Z. The exact sequence (13.63) then implies the following

isomorphism: 7rj(M)

7rj(P(N))

,

j >_ 3.

(13.64)

Next, if P(N) is 1-connected (i.e. 7r1(P(N)) = 0) then the Hurewicz isomorphism theorem implies that Z = 7r1 (S1) - 7r1 (M)

is on-to and 7r2(P(N)) -- H2(P(N); Z).

(13.65)

If dims P(N) > 2 then the Lefschetz hyperplane section theorem implies that (13.66) H2(P(N); Z) :: Z. Consequently, we find

ir2(P(N)) = Z.

(13.67)

From (13.63) and (13.67) we get the exact sequence:

0-+7r2(M)-+Z-+ Z-pir1(M)1 0.

(13.68)

Since Z -+ 7r1 (M) in (13.68) is onto, 7r1(M) is either 0 or Z. We distinguish two cases: I) 7r1(M) = 0 or II) 7r1(M) = Z.

In case I, the homomorphism Z - Z in (13.68) is also onto. Hence, it is one-to-one. Therefore 7r2(M) -+ Z in (13.68) is trivial, that is, its image is 0. On the other hand, the exactness of 0 --+ 7r2(M) -+ Z implies that 7r2(M) -. Z in (13.68) is one-to-one. Consequently, we have 7r2(M) = 0. In case II, the exact sequence (13.68) implies that Z -+ 7r1(M) is an Z is trivial. Therefore 7r2(M) --+ Z in (13.68) is isomorphism and hence Z an isomorphism. Consequently 7r2(M) = Z, and Theorem 13.9 is completely proved.

CHAPTER 13. EXTRINSIC SPHERES

212

13.4

Yano's integral formula

In this section, given a compact orientable hypersurface M"'+' in CHxn+' we establish the following integral formula: II'C{g1I2

1

2 Mzn+i

IM2,.+1

C2n21w(£)2 -

* 1 + 2nvol(M2n+1)

1 IIwI12

+trace(A2) + (2n + 1)IIHII (IIBlII

+

n 2

IIB'II2

- w(PAC)

- 71(AC))

- i(AC)IIBlII) *1

(13.69)

(the necessary definitions are given below) and refine the classification in Section 13.2 in connection with the question of whether the (almost) contact vector t; of M2n+1 is Killing. Note that (13.69) is similar to an integral formula by K. Yano & M. Kon (cf. (11.6) in [296], p. 130) in that both are based upon a Bochner-type formula obtained earlier by K. Yano, [294]. The Yano-Kon integral formula proved to be very effective in classifying compact orientable CR submanifolds (of the complex projective space) with semi-flat

normal connection and parallel f-structure. We may state (cf. (81])

Theorem 13.10 Let M2n+1 be a compact orientable totally umbilical real hypersurface of the complex Hopf manifold 1) Assume M2n+1 to be tangent to the Lee field of a) M2n+1 is totally geodesic. CHa+'.

CHa+1.

Then

b) If B and t; are orthogonal then n = 1 and t' is a Killing vector field. Conversely, let M3 C CHa ; then t; is Killing if and only if B and 1; are orthogonal.

2) Assume M2n+1 to be normal to the Lee field of Then the mean curvature vector H of M2n+1 is parallel in the normal bundle. Also t; is Killing if and only if M2"+1 is totally geodesic. If Bl + 2H = 0 and M2n+1 is connected and simply connected then M2n+1 is isometric to the standard unit sphere S2n+1 in Cn+1 yet t is not Killing. CHa+'.

It is well known that the natural metric on S2n+1 is Sasakian (cf. e.g. [25]) and in particular K-contact; yet, the induced almost contact metric structure of a totally umbilical real hypersurface M2n+1 of CHa+1 is generally not normal (e.g. it is locally conformal cosymplectic if and only if BI + 2H = 0, cf. M. Capursi & S. Dragomir, [47]) and thus the isometry M2n+1

13.4. YANO'S INTEGRAL FORMULA

213

S2n+1 (given by 2) in Theorem 13.10) does not preserve the almost contact

structure. Let N be a unit normal vector field on M2,+1. We shall make use of the Gauss and Weingarten formulae (13.1)-(13.2) of j : Men+1 C CHn\+1. Set

= -JN. Moreover, the orthogonal comAs go is Hermitian, C is tangent to plement with respect to g = j*go of RC in T(M2n+1), is J-invariant so that, Set as is well known, Men+1 becomes a generic CR submanifold of M2n+1.

Cn+l.

?I(X) = g(X, 0 T(M2n+1). Then (P, C, 77, g) is an almost contact metric strucfor any X E ture on Men+1 (in the sense of [25]). Note that FX = 77(X)N. Using (12.8) and the Gauss and Weingarten formulae (13.1)-(13.2) one obtains

VxC = PAX + 2 {w(e)X + wo(N)PX - 77(X)B}

h(X, c) = FAX + 2 {wo(N)FX - 77(X)B') for any X E be written as

T(M2n+1).

(13.70) (13.71)

As h(X, Y) = g(AX, Y)N the identity (13.71) may

wo(N)FX - i7(X)Bl = 0. Set X = . Then (as ?7(C) = 1) we obtain Bl = wo(N)N.

(13.72)

In particular JIBlI[ = wo(N). Next, we need to compute the divergence of C. Using (13.70) and trace(P) = 0, trace(PA) = 0 because P is skewsymmetric and A is symmetric, we may conduct the following calculation:

div(e) _ Eg(DEC,Ej)

_ > g (PAEI + 2 = where {El,

,

wo(N)PE; - 77(E;)B}, E,

2n +l 2

1

w(e) - 211(B)

E2n+1 } is a local orthonormal frame on M2,+ 1. Thus,

div(e) = nw(C).

(13.73)

CHAPTER 13. EXTRINSIC SPHERES

214

We recall (cf. K. Yano, [294]) the following identity: Ric(X,X) + 1 JJLXgII2 - IIVXII2 - (divX)2

= div(VxX) - div((divX)X)

(13.74)

for any X E T (M2i+1) Here C is the Lie derivative. Let us show that (13.74) leads to (13.69). A deeper application of (13.74) is demonstrated in Chapter 16 in connection with the classification of submanifolds fibered in tori of a complex Hopf manifold, cf. [11]. The Gauss equation (12.65) (of j) may be written

R(X, Y)Z = (X A Y)Z + (AX A AY)Z

+4{[w(X)Y - w(Y)X]w(Z) +[g(X, Z)w(Y) - g(Y, Z)w(X)]B}

(13.75)

for any X, Y, Z E T (M2n+1) Suitable contraction of indices in (13.75) leads to

Ric(X,Y) _ (2n - 4IIwlI2) 9(X,Y) +(2n + 1)g(AX,Y)I) HII

2n4

1W(X)w(Y)

- g(A2X,Y).

(13.76)

Set X = Y = in (13.76) to obtain 2n - 2IIwII2

- 2n4

lw(t )2

+(2n +

(13.77)

Next, we need to compute the norm of V terms of a (local) orthonormal frame {El,

.

To this end, we recall that in , E2n+1 } on Men+1 we have

IIvXII2 = >g(VE;X,E;)2.

(13.78)

IIVCII2=Sl+S2+4S3

(13.79)

Thus where

S1 = > g(PAEE, E;)2 to

215

13.4. YANO'S INTEGRAL FORMULA

S2 = E g(PAEi, Ei) {w(C)g(E.i, Ei) i,i

+ wo(N)g(PEE, Ei) - rl(EE)w(Ei)} (w(C)g(E,, Ei) + wo(N)g(PEE, Ei) - r,(Ei)g(B,

S3 =

E,))2.

i,,j

As g(PX, Y) + g(X, PY) = 0 and p2 = -I + r ®e we may perform the following calculation:

Sl = > g(PAEj, PAEj)

-

g(P2AEi, AEi) = trace(A2) - g(A2C, C).

Next

S2 = > (w(.)g(PAEj, Ei) +wo(N)g(PE3, PAEJ) - r1(E3)g(B, PAE,))

= w(C)trace(PA) - 1: wo(N)g(P2AEj,E3) i -g(B, PAC) = wo(N) (trace(A) - S(AC)) - w(PAC). Finally S3 = (2n + 1)w(e)2 +> (wo(N)2111'Ei112 - 211(Ei) [w(e)w(EE)

i

+ wo(N)w(PE,)} +

IIw11277(Ei)2) .

Note that Ei 77(Ei)2 = 11,1112 = 1 and trace(P2) = -2n. Also PC = 0. Thus, S3 = (2n - 1)w(C)2 + IIwI12 + 2nwo(N)2.

Therefore (13.79) may be written as IIV

II2 = trace(A2) - g(A2e)

+wo(N) (trace(A) - 71(AC)) - w(PAC)

+2n4

1

w(C)2 + 4IIwII2 + 2wo(N)2.

(13.80)

CHAPTER 13. EXTRINSIC SPHERES

216

Set X =

in (13.74) and use (13.73), (13.77) and (13.80) to write the

resulting identity as div(V ) = 2IIGegII2 + n.(w(C)) + 2n - 2IIwII2

2n - lw(e)2 2

+ (2n + 1)i7(A = C)IIHII - trace(A2)

-wo(N){(2n + 1)IIHII

-

2wo(N)2.

(13.81)

Assume from now on that M2n+1 is compact. Integrate in (13.81) over M2n+1 and use Green's lemma to yield (13.69), Q.E.D. At this point we may prove Theorem 13.10. Recall (cf. e.g. [2691) that IIwoII = 2. Also by Green's lemma,

f

e(w(e)) * 1 = 0.

M2n+1

(13.82)

The Codazzi equation (10.20) and the total umbilicity furnish

V*H = -1w(X)B1.

(13.83)

Assume Bl = 0. Then by Theorem 13.6 Men+1 is totally geodesic (A = 0). Then by (13.82) the integral formula (13.69) becomes 1

2

f M2n+1

IIG£9II2 * 1 + 2(n - 1)vol(M2n+1)

= 2n2 + 2n - 1 / 2

(13.84)

1.

M2n+1

If B 1 C then (by (13.84)) we get n = 1 and Gig = 0. If n = 1 then IIL4g1I2 * 1 = 3 fM3 w(e)2 * 1 fM3

yields the last statement in b). Let Men+1 C CHa+1 be a compact orientable real hypersurface normal to the Lee field of CHn +1. Then (13.69) may be written 2 JM2n+1 III

(trace(A2)+

gII2 * 1= f

+(2n + 1)IIHII(2 -

Men+1

27,(Ae))

.

(13.85)

13.4. YANO'S INTEGRAL FORMULA

217

Next, if M2n+1 is totally umbilical then on the one hand (13.83) gives V'H = 0 and on the other (13.85) becomes fM2fl+1

8nIIHII) * 1 = 0

(13.86)

i.e. a is Killing if and only if H = 0. Assume from now on that

2H + Bl = 0.

(13.87)

Then (13.86) shows that is not Killing. Indeed, if Ltg = 0 then (13.86)(13.87) would give Bo = 0, a contradiction. Finally, let us check that under the given hypothesis one has M2n+1 - S2n+1

(13.88)

(an isometry). As M2n+1 is compact, the induced metric g is complete. By (13.87) we have k = IIHII = 1 and H E D1 = RBo so that we may apply i) of Theorem 13.4 to conclude that (13.88) holds.

Chapter 14

Real hypersurfaces The purpose of this chapter is to study the geometry of real hypersurfaces of a I.c.K. manifold, in particular of a complex Hopf manifold carrying the Boothby metric.

14.1

Principal curvatures

Leaving definitions momentarily aside we may state

Theorem 14.1 Let

M2i-1

be an orientable connected real hypersurface of

the g.H.o manifold M02n , n > 3, having exactly two constant principal curva-

tures, both of multiplicity > 2. Then M2i-1 is locally isometric to a product L x L' of totally umbilical submanifolds of Mon having a flat normal connection. Moreover, either is tangent to the Lee field of Mpn, or L, L' are extrinsic spheres of Mon. Men-1

Theorem 14.2 Let M2n-1 be an orientable connected quasi-Einstein (i.e. Ric = pg-(2n-3)(w®w)/4) real hypersurface of a g.H.0 manifold Mon, n > 3, I1woII = 2c, c E (0, +oo), tangent to the Lee field of Mpn i) p< (2n - 3)c2/4.

ii) If p = (2n-3)c2/4 then either Men-1 is totally geodesic in Mon or it has exactly two principal curvatures, one nonzero of multiplicity 1, and the other zero (of multiplicity 2n - 2). iii) If p < (2n - 3)c2/4 then is locally isometric to a product of two extrinsic spheres of Mon Men-1

Cf. S. Dragomir, [77], p. 379. To prove Theorems 14.1 and 14.2, let ' :

M2n-1

Mon be an orientable real hypersurface of the g.H.o manifold

CHAPTER 14. REAL HYPERSURFACES

220

M2-1 and set as usual A = AN. Let Mpn. Let N be a unit normal field on Al E C°°(M2r-1) , 1 < j < 2n - 1, be the principal curvatures of M2n-1, i.e.

Spec(A1)={Ai(x):1<j 2 and q > 2.

As p>2one has wA(X)=0 foranyXETa. Next (by (14.3) and const.) one obtains

(A - A)VyX = 0

(14.7)

wo(N)w(X) = 0

(14.8)

and

for any X E Ta,Y E Tµ. Similar to (14.7), one also has (A - p)VXY = 0. This suffices for establishing the parallelism of both T,\, T.. For instance, if

Z E Ta then (as Vg = 0) one has 0 = g(VZX,Y) + g(X,VZY) and thus VZX E TA,, etc.

CHAPTER 14. REAL HYPERSURFACES

222

Remark 14.2 Bl has singular points. The proof is by contradiction. Suppose Bz # 0 for any x E M2,-I. Then (14.8) yields w = 0 on TA. Since (similar to (14.8)) one may obtain wo(N)w(Y) = 0 for any Y E Tµ, it is clear that w - 0. This has the following geometric meaning. Let Fo be the canonical foliation of Mo" (cf. Chapter 5) whose leaves are the maximal connected integral manifolds of the Pfaffian equation wo = 0. Thus Men-1

appears to be a leaf of Fo. Consequently M2i-1 would be, by (10.39), totally geodesic in Mon and therefore A = p = 0, a contradiction.

Step 4. The following formula holds:

p4(AFt+ 4) =

t,,B+fcµBII2.

(14.9)

Let X E Ta, Y E T. By Step 3, To is parallel, so that R(X, Y)Y E To.

Thus g(R(X, Y)Y, X) = 0.

(14.10)

On the other hand, by the Gauss equation (10.88), one obtains R(X, Y)Y = (Ap + )IIYII2X

+4{Iw(X)Y - w(Y)X]w(Y) - IIYII2w(X)B}.

(14.11)

Take the inner product of (14.10). This procedure leads to

(Ap+ )IIX12 11Y112 -

4{w(Y)211XII2+w(X)2IIYII2} = 0.

(14.12)

Let {Et}1 2. Now t(x) = dimR(Im(Ax)). Consequently there exists Y E Tx(M2n-1) such that AxY 0 and (AxX, AxY) = 0, where (,) = gx. As X E So(x) one obtains (by (11.75)) AxX A AxY = 0 and then IIAxX112AxY=0

i.e. X E Ker(Ax). We conclude that So(x) = Ker(A1)

(14.20)

M2n-1, y # x. As M21-1 is homogeprovided that t(x) > 2. Let y E neous, let f E Isom(M2n-1) such that f(x) = y and (dxf)Bx = B. Then

(dx f)So(x) C So(y). Consequently dimR So(x) < dimR So(y). The reversed inequality may be proved in a similar way. Thus t(x) = t(y). So, if t(x) > 2 then t(x) = const. If, in turn t(x) < 1 for some x E Men-1 then (by (11.75)) one obtains dimR So(x) = 2n - 1. Thus, by the homogeneity assumption, dims So (y) = 2n - 1, for all y E Men-1. Using again the Gauss equation Men-1, Q.E.D. (11.75) it follows that rank(A) < 1 everywhere on

14.4

Type numbers

The goal of this section is to report on the following results (cf. M. Capursi & S. Dragomir, [49]):

14.4. TYPE NUMBERS

227

Theorem 14.4 Let M2i-1 be an orientable real hypersurface of the complex Hopf manifold CH) n,. If M2,-1 is tangent to the Lee field of CHa and the f structure P of M2i-1 anti-commutes with the Weingarten operator A, then M2i-1 is mixed foliate and at any point of M2r-1 its type number is < 1. Theorem 14.5 Let M2r-1 be an orientable connected real hypersurface tangent to the Lee field of CHa . Suppose that the structure vector = -JN is an eigenvector of the Weingarten operator A corresponding to the eigenvalue

a E C°°(M2n-1). If t(x) > 2 for all x E M2r-1 then either a - 0 or a is nowhere vanishing and then w is exact, i.e. w = -d log a2.

Let us recall (cf. e.g. [296], p. 93) that a CR submanifold (Mm, D) of a Hermitian manifold Mon is mixed foliate if it is mixed totally geodesic and

h(PX, Y) = h(PX, Y) for any X, Y E D. We have (cf. S. Dragomir, [76], p. 2)

Theorem 14.6 Let Mm be a mixed foliate generic CR submanifold of the complex Hopf manifold CHa . If Mm is tangent to the Lee field of CHa then Mm is D-geodesic. Moreover D is integrable and its leaves are totally umbilical submanifolds of Mm.

Combining Theorems 14.4 and 14.6 we obtain

Corollary 14.1 Let M2n-1 C CHa be an orientable real hypersurface such that Bl = 0 and AP + PA = 0. Then M2n-1 is D-geodesic and foliated by complex manifolds each of which is totally umbilical in

M2n-1.

We shall need the following:

Lemma 14.3 Let M2n-1 C CHa be an orientable real hypersurface. Then

OxC = PAX + 2{wo(N)PX +w(f)X - i7(X)B} for any X E

(14.21)

T(M2n-1).

Proof. To compute the covariant derivative of l; one uses the Gauss formula (11.1) and (10.8). M2n-1 C CHa be an orientable real hypersurface and suppose Let

from now on that t; is an eigenvector of A, that is, At; = at; for some

CHAPTER 14. REAL HYPERSURFACES

228

a E C°°(M2i-1). By (14.21) we obtain

(VxA)e = X(a) +aPAX -2{wo(N)IAPX - aPXJ +w(E)[AX - aXJ -77(X)[AB - aBJ}

(14.22)

for any X E T(M2"-1). Now (14.22) and

g((VxA)Y,.) = g(Y, yield

g((VxA)Y, C)

= X(o)ii(Y) + ag(PAX, Y) - g(APAX, Y) -2{wo(N)[g(APX,Y) - ag(PX,Y)J +w(C)[g(AX,Y) - ag(X, Y)] -7)(X)[w(AY) - ew(Y)J}

(14.23)

for any X, Y E T(M2n-1). Furthermore (14.23) and the Codazzi equation (12.5) lead to 4{w(Y)7I(X)

- w(X)n(Y)}wo(N) = X(a)i7(Y) -Y(a)n(X) -wo(N)[2g((AP + PA)X,Y) - ag(PX, Y)J +ag((AP + PA)X, Y) - 2g(APAX, Y) +2{7j(X)Iw(AY) - aw(Y)J

-77(Y)[w(AX) - aw(X)J}.

(14.24)

Let us put Y = in (14.24). We obtain X(a) = Va)77(X) + 2[w(AX) - aw(X)J w(X)Jwo(N)

(14.25)

for any X E T(M2s-1). At this point we may substitute from (14.25) into (14.24) to yield

g(LX,Y)[a - 2wo(N)J = awo(N)Sl(X,Y) - 212(AX,AY)

(14.26)

14.4. TYPE NUMBERS

229

where

L = AP + PA. Now we may prove our Theorem 14.4. If L = 0 then PAC = 0 gives

AC=rl(AC) i.e. C is an eigenvector of A with the eigenvalue a = 77(Ae). Thus, under the hypothesis of Theorem 14.4, we may use (14.26). Let us consider (14.26) for

Y = PX (and L = 0, a = 77(Ae)). We obtain 2I1APX II2 + r7(Ae)I]PX []2wo(N) = 0.

(14.27)

Men-1, Therefore, if B = 0 then A = 0 on the holomorphic distribution V of and Theorem 14.4 is proved. Let M2r-1 C CHa be an orientable real hypersurface with AC = ae for C°O(M2n-1) and set 3 = C(a). Our next step is to compute the some a E covariant derivative of the gradient (da)ft. Note that (14.25) reads

da = 077 + 2 (w o A - aw) + (w(e)77 - w)wo(N). 4

Since 770 =

(14.28)

and wO = B, applying the isomorphism # to (14.28) gives

(dcr)a = QC + 2 [AB

- aB] +

4

B]wo(N).

(14.29)

Take the covariant derivative of both members of (14.29) and use the identities VxB = wo(N)AX , X(wo(N)) = -w(AX) together with (14.21) to obtain

Vx(da)d

= X (,3)C - [a + 2wo(N)]wo(N)AX

-14 [w(e)e - B]w(AX) +2I [VxAB - X(a)B] +4X (w(C))wo(N)C + [,6 + 4wo(N)]{PAX

+1 [wo(N)PX +w(e)X -17(X)B]}.

(14.30)

Set f = wo(N) for simplicity. Since the Lee form of CH' is parall el, the Gauss formula (11.1) furnishes

(Vxw)Y = g(AX,Y) f.

(14.31)

CHAPTER 14. REAL HYPERSURFACES

230

Now (14.31) and Lemma 14.3 give

X(w(e)) = [71(AX) + 1w(PX)]f +w(PAX) +2[w(Ow(X) - i1(X)Ilw1121-

(14.32)

Finally, we may substitute from (14.30) to obtain the following:

Lemma 14.4 Let M2"-1 C CH' be an orientable real hypersurface with Al; = al; for some a E C°O(M2s-1). Then V x (da)d = X (#)t; + /3PAX +2 f {,OPX + 1w(1;)PAX 2

aAX + 1W(PAX)1; 2

+4[w(C)w(X)-ii(X)Ik +4 f2{[ii(AX) + 1w(PX)]k - AX + 2w(£)PX)

+2 {Q[w(C)X - 71(X)B] -1 [w(C)l; - B]w(AX) + VxAB - X(a)B}

for any X E

(14.33)

T(M2n-1)

Now we may prove Theorem 14.5. Assume f = 0. Then

woA=O , A(B)=0 Vx(da)d =X(/3)1;+,3PAX -

2{[X(a)+J3r1(X)]B-,Qw(1;)X}.

(14.34) (14.35)

Clearly

(Vxda)Y = (Vyda)X because V is torsion-free. Thus (14.35) gives

Y(P)77(X)-X(fl)77(Y) = 6[g(PAX,Y) - g(PAY, X)1 + 2 {Q[w(X)77(1') - w(Y)71(X )1

+w(X)Y(a) -w(Y)X(a)}.

(14.36)

14.4. TYPE NUMBERS

Let us put Y =

231

in (14.36). We obtain

XW) = c(Q)77(X) -,8w(X) +2w(£)[X(a) +,(377(X)].

(14.37)

Now substitution from (14.37) into (14.36) yields

,6L = Z,O[w®r7(9 B] +2{(da) ® [B -

[w -

(da)p}.

(14.38)

As f = 0 our identities (14.29)-(14.30) reduce to 1

da =Q-taw (da)p = Qe

- 2 aB

so that (14.38) gives the following expression of the operator L: 4/3L = a[w ®£ - r7 ®

(14.39)

Again by f = 0 our (14.26) gives ag(LX, Y) = 2g(PAX, AY) or

g(LX, aY) = g(APX, AY)

(14.40)

where

a = A -aI. Let us take (14.40) at Y = PX and substitute L from (14.39). We obtain 8QIIAPXII2 = a 2w(C)rl(X)w(PX)

(14.41)

for any X E T(M2i-1). Now since L£ = 0 the identity (14.39) gives a[w(C) - B]w(C) = 0.

Applying P to this identity we have

aw(f)PB = 0.

(14.42)

CHAPTER 14. REAL HYPERSURFACES

232

Let x E Men-1. We distinguish two possibilities. Either (PB)x # 0 and thus (by (14.42)) we have a(x)w(e).., = 0 which yields (by (14.39)) ,Q(x)Lx = 0.

If this is the case we obtain /3(x) = 0 or Lx = 0. But Lx = 0 would give Ax = 0 on Dx (by repeating the proof of Theorem 14.4), i.e. t(x) < 1, a contradiction, such that we are left with /3(x) = 0. Or, the second possibility is (PB)x = 0 and then w(PX)x = 0 and (14.41) implies,6(x)DDAPXIjx = 0 with the consequence /3(x) = 0 or Ax(PX )x = 0. The last equality yields t(x) _< 1, a contradiction, so that again we are left with /3(x) = 0. Thus /3 - 0 and (14.28) gives da + (1/2)aw = 0. Consequently, we have (locally) as = cont., for some nowhere vanishing function A. Since M2t-1 is connected we

conclude that either a - 0 or a(x) # 0 at any x E M2n-1, Q.E.D.

14.5

L. c. cosymplectic metrics

In this section we describe the geometric structure inherited by real hypersurfaces from their l.c.K. ambient space. This turns out to be a locally conformal cosymplectic (l.c.c.) structure (in the terminology of Z. Olszak, 1211]). We indicate the line which may be followed to further study these structures (cf. M. Capursi & S. Dragomir, [47J) in Section 17.6. We start with the following general considerations. Let

(M"'1,W,C77,9) be an almost contact metric (a.ct.m.) manifold of real dimension 2n + 1 (cf. D.E. Blair, [25], p. 19-20). It is said to be normal if N1 = 0 where

N'

= [co, caJ + 2(dn) ®l;.

An a.ct.m. manifold is cosymplectic if it is normal and both the almost contact form 77 and the fundamental 2-form 9(X, Y) = g(X, spY) are closed (i.e. dr1 = 0 and de = 0). See D.E. Blair, [26], Z. Olszak, [210], S. Tanno, [250], for the general properties of cosymplectic manifolds. Within the class of a.ct.m. manifolds, Sasakian manifolds have received full attention over the last thirty years, while cosymplectic manifolds were comparatively less extensively studied. Cosymplectic manifolds appear to be the closest odddimensional analogues of Kahlerian manifolds and several known results from

Kahlerian geometry carry over to the cosymplectic realm often with only mild alterations of the original proofs. Nevertheless, there are differences (which ought to be explored) among Kiihlerian and cosymplectic geometry. D. Chinea & M. De Leon & J.C. Marrero, [67], focus on these differences and

14.5. L. C. COSYMPLECTIC METRICS

233

ultimately Caesari quod Caesaris est reddit. For instance, the Betti numbers of a compact cosymplectic manifold M2n+1 are shown to satisfy bo < bl < . < bn = bn+1 ? bn+2 ?

? b2n+1

thus refining a result of D.E. Blair & S.I. Goldberg, [29], and filling in a gap of the proof therein. Again, see [67] for a cosymplectic analogue of the Lefschetz theorem in Kahlerian geometry.

Let M2n+i be an a.ct.m. manifold. Then M2n+1 is said to be locally conformal cosymplectic (l.c.c.) if there exists an open covering {Ui}iEf of M2n+1 and a family {fl}iEJ, fi E C°°(Ui), of R-valued smooth functions

such that (Ui"Pi, Si, rli, 9i)

is a cosymplectic manifold, where exp(

cci = WIU,

9i = exp(- 2t )i7,u,

,

gi = exp(-fi)9iu,

for any i E I. Clearly, if M2n+1 is I.c.c., then cc is integrable. Let M2n+1 be an a.ct.m. manifold and f E C°O(M2n+1) a smooth R valued function on M2n+l. A conformal change of the a.ct.m. structure (cf. I. Vaisman, [280]) is a transformation of the form

Wf = cP , (f = exp(2 ) , 77f = exp(-

)'9

, 9f = exp(-f)9.

(14.43)

2 The Riemannian connections of g, g f are related by

V Y = OXY - a [X (f )Y + Y(f)X - 9(X, Y)grad(f)) where grad(f) = (df )l and d denotes raising of indices with respect to g. Clearly

(M2n+1,

cP, f, n]f, 9f)

is an a.ct.m. manifold and is cosymplectic if and only if d77 =

2 df A 77

de =dfn6 [cP, cP) = 0

where A(X, Y) = g(X, ccY). Then

CHAPTER 14. REAL HYPERSURFACES

234

Lemma 14.5 Let (M2n+1,

, i, g) be a cosymplectic manifold, n > 1. If the cosymplectic property is invariant under the transformation (14.43) then df = 0 on M2n+1 W,.

Consequently one may establish (cf. Theorem 5 in [47], p. 31) Theorem 14.7 Let (M2n+1, gyp, , 77, g) be a 1. c. c. manifold. Then for any

i, j E 1, i # j, with U; fl U3 # 0, one has dfi = df3

on Ui fl U,. Therefore the (local) forms dfi glue up to a globally defined (closed) 1-form w. Also the Riemannian connections V1 of (Ui,gi) glue up to a globally defined linear connection D on M2n+1 expressed by

DXY = OXY - 2 [w(X )Y + w(Y)X - g(X, Y)B] where B = wa and V is the Levi-Civita connection of (M2n+1,g)

The 1-form w is referred to as the characteristic 1 form of M2n+1 Also B is the characteristic field and D the Weyl connection. Since dtti = 0, dOi = 0, where Oi denotes the fundamental 2-form of (gyp, li, iii, gi), it follows that d77=

1wnil , dO=wAO.

(14.44)

Also, for any l.c.c. manifold kP, W1 = 0.

Conversely, any a.ct.m. manifofold M2n+1 satisfying (14.44) for some closed 1-form w and with cp integrable is l.c.c. If w =- 0 then M2n+1 is a cosymplectic manifold. If Sing(w) = 0 then M2n+1 is called strongly non-cosymplectic.

The reason we bring the l.c.c. structures into the picture may be explained as follows. Let (Mon+2, J, go) be a l.c.K. manifold. Let M2n+1 be Let N be a unit normal field an orientable real hypersurface in M02n+2.

and set l; = -JN. Also, set cpX = tan(JX), FX = nor(JX), for any Previously, the f-structure cp was denoted by P; while here XE we wish to unify our notation with that usually adopted in contact geometry, cf. e.g. [25]. Let E(c) be the normal bundle of t : M2n+1 C Mon+z Let T(M2n+1).

i7(X) = go(FX, N).

14.5. L. C. COSYMPLECTIC METRICS

235

By a result of [25], p. 30, (cp, l;, rl, g) is an a.ct.m. structure on Let wo be the Lee form of M2n+2 and w = t*wo. If Ho is the Kithler form of (M2n+2, j, go) then set A = t*11o. Then M2n+1.

d9=wA9

,

dw=0.

We may state (cf. [47], p. 34)

Theorem 14.8 Let M2n+1 be an orientable real hypersurface of the 1.c.K. manifold M02n+2 and assume that either M2n+1 is totally umbilical and its mean curvature vector satisfies

H=-1Bl

Bl=nor(Bo)

,

or M2n+1 is totally geodesic and tangent to the Lee field Bo of M02n+2 Then (cp, , tl, g) is a 1. c. c. structure on M2n+t

To close this section, we look at certain submanifolds of complex Inoue surfaces carrying the Tricerri metric, cf. Section 3.2. Let C+ = {z E C : Im(z) > 0} be the upper half of the complex plane. Let (z, w) be the natural complex coordinates on C+ x C. We endow C+ x C with the Hermitian metric

Go = y-2dz ®dz + ydw ®dw

where z = x + iy. This makes C+ x C into a globally conformal Kahler manifold with the Lee form wo = y-1dy.

Let A E SL(3, Z) with a real eigenvalue a > 0 and two complex eigenvalues

/3 0,3. Let (at, a2, a3)

,

(b1, b2, b3)

be a real eigenvector and an eigenvector correponding to a, /3, respectively.

Let GA be the discrete group generated by the transformations f., a = 0, 1, 2, 3, where fo(z, w) = (az, /3w) and fi(z, w) = (z + a;, w + bi), i = 1, 2, 3. Then GA acts freely and properly discontinuously on C+ x C so that SA = (C+ X C)/GA becomes a (compact) complex surface. This is the Inoue

surface, cf. Chapter 3. It was observed (cf again Section 3.2) that Go is GA-invariant. Thus Go induces a 1.c.K. metric go on SA. Cf. Prop. 2.4 in [258], p. 85, the corresponding Lee form is not parallel so that SA with this l.c.K. structure is not a g.H. manifold.

CHAPTER 14. REAL HYPERSURFACES

236

Let A : C+ x C -+ SA be the natural covering map. Let t : M C C+ x C be a submanifold and g = t*Go. Then' : M -+ SA, 'I' = Trot, is an isometric immersion of (M, g) into (SA, go). It is our purpose to build examples of (immersed) submanifolds of SA and motivate the results in Section 14.6. Let w = u + iv. We set

a 8a

X

'Y

ay

a

i9

19

au,V

,U

(9v

The real components of go are

y-2 0

0

0

0

0

0

y 0

0

0

0

y

0

go :

y-2 0

Thus the non-zero Christoffel symbols of the Levi-Civita connection V° of (SA, go) are:

r13 = I33 - -r11 1 r23=r34=2y-1

-y-1

1

r22=r44=_2y2

The Lee field of SA is (locally) given by

Bo=yY. Let

Lh = {z E C+ : Im(z) = 1}

and let t : Lh X C C C+ X C. The tangent space at a point of Lh X C is spanned by X, U and V. Then N = Bo is a normal vector field on Lh X C. By looking at the Christoffel symbols computed above one gets

VON=-X, VON= 2U, Vov= 2V. Let aN be the shape operator of W : Lh X C -+ SA, 4< = a o t. Then trace(aN) = 0, i.e. T is minimal. Clearly Lh x C is a maximal connected integral manifold of the Pfaffian equation y-'dy = 0 (i.e. a leaf of the canonical foliation of the strongly non-Kahler l.c.K. manifold SA).

LetL°={zEC+:Re(z)=0}andt:L°xCCC+xC. Tangent

spaces at points of L' x C are spanned by Y, U, V, and N = yX is a normal vector field. Then DUU =

-

21

y2Y

, vUY =

2y-lU

,

vUV = 0

14.5. L. C. COSYMPLECTIC METRICS

237

V Y = -y-lY , V°}.V = 2y-1V , V°VV = -1y2Y. Consequently T : L" x C

SA, T = r o t, is a totally geodesic immersion.

Clearly L' x C is tangent to B0 and inherits a l.c.c. structure (via our Theorem 14.8). Both Lh x C and L° x C are generic, as real hypersurfaces

of S.

Chapter 15

Complex submanifolds The study of complex submanifolds of a l.c.K. manifold has been initiated by I. Vaisman, [275], see Theorem 12.1. A further investigation, regarding complex hypersurfaces of a l.c.K. manifold, was made by S. Ianu§ & K. Matsumoto & L. Ornea, [133]. Complex hypersurfaces of a Vaisman manifold with planar geodesics were studied by L. Ornea, [217]. Clearly, any complex

hypersurface M of a 1.c.K. manifold Mp" is itself l.c.K. when tangent to the Lee vector field. If Mo" is a Vaisman manifold then, by (12.39), M is a Vaisman manifold if and only if ABl = 0. In particular, any complex hypersurface tangent to the Lee field of a g.H. manifold is a g.H. manifold, too. Cf. Prop. 2.1 in [133], p. 124. Complex submanifolds (of a l.c.K. manifold) with additional restrictions on curvature have been studied by S. Dragomir, [75] (cf. Theorems 12.13, 12.15, 12.16 and 12.20). As a natural deepening of these results we report on [82].

15.1

Quasi-Einstein submanifolds

The study in [133] is confined to the case of complex hypersurfaces of a Vaisman manifold with B-L = 0. By Theorem 12.1 these are minimal and an inspection of the expression of their Ricci curvature following from the Gauss equation (12.88) and H = 0 shows that there are no Einstein complex hypersurfaces with Bl = 0 in a g.H.o manifold. Cf. Corollary 4.2 in [133], p. 127. The verification is similar to the proof of ii) in Theorem 12.15. Also, we may state

Theorem 15.1 Let M be a complex hypersurface of a g.H.o manifold with Bl = 0. Then M is totally geodesic if and only if Rl = 0.

CHAPTER 15. COMPLEX SUBMANIPOLDS

240

Cf. [133], p. 127. Let M be a complex hypersurface of a g.H.o manifold.

If B-'- = 0 and Rl = 0 then (by Theorem 15.1) an inspection of the Ricci tensor expression (or Theorem 12.13) shows that M is quasi-Einstein. As to the arbitrary complex codimension case, we obtain (cf. [76])

Theorem 15.2 Let M21 be a complex submanifold of a [/airman manifold Mpn. If M2s has a flat normal connection (R1 = 0) and is tangent to the Lee field (Bj- = 0) of Mpn then the Ricci tensors Ric and Rico, of (M2,, g) and (Mon, go) respectively, satisfy

Rico(X, Y) = Ric(X,Y) + 2(n - s){czg(X,Y) - 4w(X)w(Y)}

(15.1)

for any X, Y E T(M2s), where ]]wI] = 2c. Thus, if Mon is quasi-Einstein (i.e. Rico = ago + bwo ® wo for some a, b E Coo(Mpn)) then M2' is quasiEinstein, too. Assume b = coast. and n > 2. Then M2' and M02' have the same (constant) scalar curvature if and only if a + (2s - 1)c2 = 0. We shall prove Theorem 15.2 later on. Cf. Theorem 4.4 in [133], p. 127, any complex hypersurface, with Bl = 0, in a g.H.o manifold, having a parallel

second fundamental form, is locally symmetric. This is a corollary of the following general result:

Theorem 15.3 Let Mm be a submanifold of the g.H.o manifold Mpn. If m > 2 and Sing(w) = 0 and MI has a parallel second fundamental form, then MI is locally symmetric. Cf.

Theorem 7 in [76], p. 7. This is proved by standard arguments by

observing that any g.H.o manifold is (by (12.13) with Ko = 0 and V°w° = 0) actually a locally symmetric Riemannian manifold (i.e. V°Ro = 0). Let us prove Theorem 15.2. Let M2s be a complex submanifold of the

g.H. manifold Mon. As Bl = 0 it follows by (12.39), that Ow = 0. As R' = 0 there is (cf. e.g. Prop. 1.1 in [52], p. 99) a global orthonormal , £n_ JC1, , Jt n_,} of E(T) so that V' 0 = 0. As frame {CI,

= Jvx

VX

the remaining fields JC are parallel, as well. Let {E1, orthonormal frame of T(M2s). Recall that 2s

Ric(X, Y) _

g(R(Ei, X )Y, E;), i=1

,

EE,} be a local

15.1. QUASI-EINSTEIN SUBMANIFOLDS

241

etc. Then by the Gauss equation of T, 2s

Rico(X,Y) = Ric(X,Y) +>2go(h(E1,X),h(Ei,Y)) i=1 n-s

+ T{Ro(ea,X;Y,&a)+Ro(Je,,,X;Y,JE.)}

(15.2)

a=1

because of wo(h(X, Y)) = 0. Here the Riemann-Christoffel 4-tensor of Mpn is given by Ro(X, Y; Z, W) = go(Ro(X, Y)Z, W). Set

Aa=&, 1 2 then a = const. and p = const. the manifold is always assumed to be connected. These considerations may

be applied to prove the last statement in Theorem 15.2, as follows. As Rico = ago + bwo ® wo it follows that r

{Rico(Ca, a) + Rico(J(a, JEa)} = tar. a=1

Also p = trace(Ric) and (15.1) yield po = p + 2(n - s)(a + (2s - 1)c2).

Finally b = const. and n > 2 yield a = const. and po = const. Thus p = const., etc.

CHAPTER 15. COMPLEX SUBMANIFOLDS

244

15.2

The normal bundle

If go is a Kahlerian metric and' : M Mo" a given holomorphic immersion

with Rl = 0 then Rico(X,Y) = Ric(X,Y)

(15.11)

for any X, Y E T (M), by a result of B.Y. Chen & H.S. Lue, [58]. If M has complex codimension one (n = s + 1) in M02" then the converse holds, i.e. (15.11) yields Rl = 0 (cf. B. Smyth, [247]). It is a natural question whether this remains true in arbitary (complex) codimension. By a result of B.Y. Chen & H. Lue, [58], (15.11) yields cl(E(WY)) = 0 (and cl(E(W)) = 0 Rl = 0 provided that (MO', go) is locally Euclidean). Here cl (E(W )) E

H2(M; R) is the first Chern class of the normal bundle E(W). As to the case of holomorphic immersions of complex manifolds in an ambient l.c.K. manifold, we may ask whether (15.1) yields cl(E(WY)) = 0. While this turns out to be true for a class of complex submanifolds of a g.H. manifold (cf.

Theorem 15.4), the problem whether cl(E(W)) = 0 Rl = 0 is more subtle. Let Ro be the curvature of the ambient metric go. When go is Kahlerian the assumptions Ro = 0 and cl(E(T)) = 0 yield

cl(T(M)) = 0

(15.12)

and thus the Ricci form ry of g is exact, i.e. -y = dA for some 1-form A on M.

Thus one may apply A = t(Q) and integrate over M assuming that A has compact support to show that T is totally geodesic and therefore R1 = 0 by the Ricci equation, cf. [58], p. 554). Going back to the l.c.K. case, it appears that the complex structure of Ma" is not parallel with respect to the Levi-Civita connection of go and thus cl(T(MM") may not be expressed in terms of Ro. Nevertheless, M02" possesses the remarkable Weyl connection D°, and the assumption that D° is flat satisfied for instance if Mon = CHn

with the Boothby metric go together with cl(E(W)) = 0 once again yield (15.12). Let D be the Weyl connection of M and let K be its curvature. Let k(X, Y) = trace{Z H K(Z, X)Y} and

a(X,Y) = 4 -k(JX,Y). Then cl(T(M)) = [a] so that (15.12) yields

a=di7 for some 1-form 77 on M. We may state the following:

(15.13)

15.2. THE NORMAL BUNDLE

245

Theorem 15.4 Let M be a complex submanifold of the Vaisman manifold Mon. Then: 1) If (15.1) holds then cl (E(W)) = 0. 2) Assume that i) the Weyl connection of M02n is flat, and ii) there is a compact 1-form 77 on M, of compact support, satisfying (15.13) and such that fm a(w)Cr7 * 1 < 0. Then R1 = 0. Cf. A. Weil, [289], for the operator C. For the proof of Theorem 15.4, among other ingredients, we use a commutation formula of I. Vaisman, [272]. We also establish

Theorem 15.5 Let Mn be a complex hypersurface of CI+1 and M a complex submanifold of Mn, tangent to its Lee field. Assume that there is a parallel normal section £ E E(4 m + 1 is performed in [85]. Let Mm be a complex submanifold (m = 2s) of the g.H. manifold M02"

Then (12.21) and (12.24) hold. Therefore, the mean curvature vector H of 'F is expressed by (12.27). Hence, if %F is minimal then llwl[ = 2c. Let us prove Theorem 15.6. To this end, let gi=W*go,i

,

iEI.

Here {go,i }iE j are the local Kahlerian metrics of M02". Then D (given by (12.43)) is the Levi-Civita connection of the local metrics gi. By (12.44) and by the Gauss equation (of M n U; in (Ui,go,i)) one has

gi (K (X, Y) Z, W) = go,i(h'(X,W),h'(Y,Z)) -go,i(h'(X,Z),h'(Y,W)) (15.29)

provided that Ko = 0. It is tacitly supposed that T : M -' Mp'i is a regular embedding, so that M n Ui is open in M. As s > 2 (indeed, if s = 1 then dSl = 0 and g is Kahlerian; thus w = 0, a contradiction) one may apply Theorem 3.8 in [269], p. 277, to obtain Theorem 15.6.

Let 'F : M -p CH' be an isometric immersion of the Riemannian manifold (M, g) , dim(M) = m, in the complex Hopf manifold (with the Boothby metric). Due to I. Vaisman's theorem (i.e. Theorem 3.8 of [269], p. 277) a g.H.o manifold with llwoll = 2c will be denoted by CH"(c). Then CH' = CH' (1) (locally). If {Ea}1 n. These results have been extended by A. Gray, [119], to the case of a nearly Kiihler ambient space, and by S. Marchiafava, [176], to the case of a quaternionic Kahler ambient space. The purpose of this section is to analyse the I.c.K. and Sasakian cases. We may state (cf. L. Ornea, [218], p. 258)

CHAPTER 15. COMPLEX SUBMANIFOLDS

254

Theorem 15.9 Let Mo" be a complete l.c.K. manifold and V', W' two compact complex submanifolds tangent to the Lee vector field B0 of M0 2n. Let

M02" have nonnegative sectional curvature (k > 0) and eventually k(p) = 0 only on 2-planes p E G2 (MM") with Bo E p. If r + s > n then VI n W' t 0. Similarly, one may establish (cf. [218], p. 259) Theorem 15.10 Let Men+1 be a complete Sasakian manifold and let V', and W8 be two compact co-invariant submanifolds tangent to the structure vector field C. If M2,+' has nonnegative sectional curvature (k > 0) and

eventually k(p) = 0 only for the 2-planes p E G2(M2n+i) with 1; E p, and if

r + s > 2n+, then

V'nW'00.

The proof of both Theorems 15.9 and 15.10 closely follows the line of T. Frankel, [99]. To illustrate our ideas we sketch the proof of Theorem 15.9 (cf. [218] for a more complete discussion).

The proof is by contradiction. Suppose that V'' n W8 = 0. Then there is a unique minimal geodesic C : [0, L] --' Mo", parametrized by arc length,

which realizes the distance between V'' and W'. Moreover, C intersects both VI and W8 orthogonally (in p = C(0), respectively in q = C(L)). Let S. c Tq(MM") be the parallel translation of Tp(Vr) along C. Then dimc(SgnTq(W8)) > r+s - (n- 1) > 1. Thus there is w E S.nTq(W') with 11wli = 1. Clearly w must be the parallel translate along C of some v E TT(Vr), 1lvII = 1. Let X be the unit vector field along C obtained by the parallel translation of v (thus Xc(L) = w). By the second variation formula (cf. J. Cheeger & D.G. Ebin, [51]): L'X' (0) = g(V xX, 6')4 - g(V xX, C)p

-1 k(C, X)dt. L 0

Using (1.5) we may also get

-g(VxX,C)q+g(VxX,O)p-J 0Lk(C,JX)dt. Thus L'X (0) + L' x (0)

I

L {k(C, X) + k(C, JX) }dt < 0.

Indeed (as e is orthogonal to both Tp(V') and Tq(W')) C cannot coincide with the restriction of B0 to C, so that at least one term (among k(C, X) and k(C, JX)) is > 0. Thus at least one among L" (0) and L'jx(0) is < 0, thus contradicting the minimality of C.

15.5. PLANAR GEODESIC IMMERSIONS

15.5

255

Planar geodesic immersions

A planar geodesic immersion is an isometric immersion T : M -- Mo between two Riemannian manifolds M, Mo which maps each geodesic of M into a 2-dimensional totally geodesic submanifold of Mo. Planar geodesic immersions are closely related to isotropic immersions (in the sense of B. O'Neill, [215]). S.L. Hong, [131], and K. Sakamoto, [240], studied the case where Mo is a real space form. When Mo is a complex space form, remarkable results

were obtained by J.S. Pak, [224]. Adapting the methods in [224], L. Ornea has obtained the following:

Proposition 15.1 (Cf. [217]) Let iY : M Mo be a complex hypersurface of a Vaisman manifold Mo with flat local Kahler metrics. If is a planar geodesic immersion then there is a C°° function A : M - R so that IIh(X,X)II2 = A2 for any X E X(M) with IIXII = 1 (i.e. ' is isotropic). If in addition M is tangent to the Lee field of Mo then M has a parallel second fundamental form. Consequently, either M is totally geodesic or its shape operator A has precisely three distinct and constant eigenvalues 14, 0

and -µ. Consider the distributions

To(x) = {T-,(M): AX = 0}

T±(x) = {X E TT(M) : AX = ±µX}.

Then To, T+ are integrable C°O distributions on M. Using this fact and Proposition 14.1 one may establish

Theorem 15.11 (Cf. [2171) Let M be a complex hypersurface with planar geodesics tangent to the Lee field of the g.H.o ambient space Mo. Then M is either totally geodesic or a Riemannian product S+ x So x S_ where So, St are leaves of To, T±, respectively.

Chapter 16

Integral formulae U. Hang Ki, J.S. Pak, and Y.H. Kim have classified (cf. [1551) complete generic submanifolds, of the complex projective space, with a flat normal connection and parallel mean curvature vector, provided that the natural 1structure of the submanifold (cf. Section 14.4) and the Weingarten operator, at any normal direction, commute. Cf. also [299].

On the other hand, although not so effective results are available as yet, considerable attention has been devoted in the 1990s to the study of the geometry of the second fundamental form of submanifolds of a l.c.K. manifold, in particular of a complex Hopf manifold carrying the Boothby metric, cf. J.L. Cabrerizo & M.F. Andres, [46], S. Ianu§ & K. Matsumoto & L. Ornea, [133], [134], K. Matsumoto, [184], [185], and the authors, cf. [216], [217], and [75], [76], [77], [78], etc. Many of the results in these papers were described in Chapters 12-15. The theory of l.c.K. metrics is itself quite new and the authors reported on its main achievements in Chapters 1-11. In this chapter, we report on a result of E. Barletta & S. Dragomir, [11] (cf. Theorem 16.3). There, using an integral formula of K. Yano, (294], one classifies all compact minimal CR submanifolds, generically embedded in a complex Hopf manifold, which are fibered in tori, and have a flat normal connection, and a second fundamental form of constant length.

16.1

Hopf fibrations

Let A E C, 0 < Al J< 1. Let CHa = (C" - {01)/A,\ be the complex Hopf manifold (of generator A). Let it : C" - {0} -i CHa be the natural covering map. Let D be the lattice given by D = {27ria + (log A)b: a, b E Z}

CHAPTER 16. INTEGRAL FORMULAE

258

(where log .\ E {log IAI + i(2k7r + arg(A)) : k E Z}). Here i = vf-_11. Set

Tc = C/D. There is a natural principal TC-bundle p : CH, -+ CPi-1 (the Hopf fibration). Our starting point is the following example (of a class of submanifolds in Let mj E Z be odd numbers, mj > 0, 1 0, 1 < j < k, be positive numbers that 7? 1 r? = 1 and set

(r1) x ... x

Mml,...,mk =

(16.1)

Throughout, Sm(r) denotes the rn-sphere of radius r and center the origin in Rm+1 Also S"1 = S(1). Cf. K. Yano & M. Kon, [296J, p. 59-60, there is a generic embedding of Mml; ,mk into Sn+k (with parallel second fundamental form and a flat normal connection). Set Nmi,...,mk = p(Mml,...,mk)

where

(16.2)

Cp(n+k-l)/2

p : Sn+k

is the natural map (called the Hopf fibration as well). We shall also need the maps po : C(n+k+l)/2 _ {0} - Cp(n+k-l)/2 and 90 :

C(n+k+l)/2 - {0} -4 Sn+k , qo(z) =

Izl

Consider the commutative diagram Vml,...,mk

= t*CH(n+k+l)/2 A

Nml

I

CH(n+k+l)/2

Cp(n+k-l)/2

where

TC -.

Vml,...,mk - Nml,...,mk

is the pullback bundle of TC -4 CH(n+k+1)/2 --+ Cp(n+k-l)/2

16.1. HOPF FIBRATIONS

259

CP(n+k-1)/2 (the inclusion).

by c : Nm,,...,mk -

Note that

Vm1,...,mk = p-1(Nm,,...,mk)

(16.3)

as set. Assume for the rest of this chapter that A E R, 0 < A < 1. There is a natural map CHan+k+l)/2

q:

,n+k

,

q(ir(z)) = q0(z)

which descends to a map Vm,,...,mk - Mml,...,mk. Indeed, if 7(z) E Vm,;,mk then p(ir(z)) E Nm,,...,mk so that p(() = p(ir(z)) = po(z) = p(qo(z)) = p(z/Izl)

for some C E Mm,

... mk.

Then z = IzIe'B( for some 0 E [0,27r), hence

q(7r(z)) = e'o( E M

(as Mm,,...,mk --+ Nm, ,...,mk is a principal S1bundle). We shall need the following: Lemma 16.1 Tr(Mm...... mk) C Vm,,...,mk

Proof. Let z E Mm, ... mk. Then jzj = 1 so that p(ir(z)) = po(z) = p(qo(z)) = p(z) E Nm,... mk.

Q.E.D. Lemma 16.2 1r(Mm,....,mk) C 7r(UmEZ Sn+k(Am))

In particular, the inclusion in Lemma 16.1 is strict. Proof. Let z E C(n+k+l)/2 - {0}. Then 7r(z) E 7r(Mm,,...,mk) yields z = .1'n( for some ( E Mm,,...,mk and m E Z. Yet JCJ = 1 hence z E S,+k (Am). To prove the second statement in Lemma 16.2, let ( = ((1, , (k) where C1 = (r3, 0,

,

0) E Smi (rl). It follows that ir(f ) E

7r

Indeed p(ir(\/AC)) = po(VC) = po(C) = p(C) E Nm,..... mk, that is, ir(/X) E Vml..... mk. Assume that E 7r(Mm,..... mk). Then,

by the inclusion in Lemma 16.2, there is m E Z with I/(I = Am, a contradiction.

Let (xj) be the natural coordinates on Rnzj+1 , 1 < a < mj + 1, and set m3+ 1

u1=

1: a=1

1<jg(aQX,Y).

(16.59)

a

Here (n+2)H = trace(hM). Also as = ava and {V,} is a (local) orthonormal

frame of v(M). Assume from now on that M is minimal (H = 0). Then (16.59) for X = Y = tMV gives, as w o tM = 0 Ric(tMV, t1V) = nlltMVII2

- Ea I1aotMVII2.

(16.60)

It is our purpose to compute the term IIVtMV[[ in (16.50). We recall that

IIVXII2 = E9(VE:X,Ej)

ij

for any X E T(M) and some (local) orthonormal frame {Ei} on M. Using (16.51) we may conduct the following calculation:

IlVtMVII2 = >{9(PMavEi,Ei)2 i.i

+9(PMavEi, E.i)9(Ei, tmV)w(Ei) + 1g(Ei, ttitV)2w(Ej)2}

_ E{IIPMavEcll2 + w(PMavEi)9(Ei,tMV) +

149(Ei,

tntV)2 IIwII2}.

i

As Pmt = -I + tMFM we have

IIPMavEEII2 = ->9(avEi,PMavEi) i

E{IlavEill2 + 9(avEi, tMFMavEE)} i

trace(a2)

- >i II FMavE:II2.

Assume from now on that M is generic (fm = 0). Then the Weingarten formula (16.15) and (16.10) yield

FMav = -hM(-,tMV)

16.3. THE MAIN RESULT

273

for any V E v(M) with VV = 0. Consequently IIFMavEill2 = i IIa,,tiVII2 and we obtain the identity IIaatMVII2.

IIPMavEiII2 = trace(av) a

i

Finally II V tMV II2

= trace(a ,) -

II aatMV II2

+w(PMaVtMV) + IItMVII2.

(16.61)

Taking into account (16.60)-(16.61), the integral formula (16.50) becomes 2

fM IIGtMvgII2 * 1 = fM

(n + 1)IIVII2 * 1.

(16.62)

On the other hand, it is straightforward that

(GtMvg)(Z,W) = g([av,PM]Z,W) - 2 {g(Z, tMV)w(W) + g(W, t,V)w(Z)}(16.63) as a consequence of (16.51). We have proved the following:

Proposition 16.2 Let M be a compact minimal generic (n + 2)-dimensional submanifold of CHa with a flat normal connection. Let V E v(M) be parallel in the normal bundle. If fM

(trace(a2)

- (n + 1) II V 112) * 1 < 0

then tMV is a Killing vector field and the commutator [av, PM] is given by (16.53).

As Rl = 0 we may choose (cf. Prop. 1.1 in [52], p. 99) a global orthonormal frame {V,,} of v(M) so that V VQ = 0. It is straightforward that II hM

112

= E trace(ac) a

CHAPTER 16. INTEGRAL FORMULAE

274

and hence (16.62) yields 0

12 jIiIzgII2

s1

Q

= JM (11

M112

- (n + 1)codim(M))

*1

(16.64)

where V. = JHZQ , ZQ E DM, and codim(M) = 2N - (n + 2). Assume that M is fibered in tori TIC over a submanifold N of the complex projective

space CPN-1. If I1hA! 12 < (n + 1)codim(M)

as in the hypothesis of Theorem 16.3, then ZQ are Killing and, by Lemma 16.8 [AQ, PNJ = 0

where AQ = At, and VQ = C' , Q E v(N). As M is generic then (by Theorem 16.1) so is N (i.e. fN = 0). In particular (by Lemma 16.6) N has a flat normal connection (Kl = 0). As M is compact and Tc -+ M -+ N locally trivial, it follows that N is compact, and in particular complete. Also the minimality of M yields, by (16.19 and by hM(B, B) = hM(A, A) = 0), the minimality of N. At this point we may apply Theorem 10.4 in [296J, p. 127, to conclude that

(an isometry), where rj = mj/(n + 1), 1 _< j < k, and mj E Z are odd numbers with m3 = n + 1. Finally, it is straightforward that Tc -+ M -+ N and Tc -+ Vm1,...,mk -+ N,n,..... m are isomorphic principal bundles. The resulting diffeomorphism M.:' Vml,...,mk,

is an isometry because both M - N and Vm1, ..,mk -+ Nm1,...,mk are Riemannian submersions. Our Theorem 16.3 is completely proved.

4

Chapter 17

Miscellanea 17.1

Parallel Ilnd fundamental form

Totally umbilical submanifolds of complex space forms have been classified by B.Y. Chen & K. Ogiue, [60], cf. Theorem 1, p. 225. Their classification relies on the earlier observation (cf. [59], Prop. 3.1, p. 260) that curvatureinvariant submanifolds of a complex space form are either holomorphic or totally real. In [79] one extends these ideas to submanifolds of Sasakian space forms and obtains the following:

Theorem 17.1 Let M2m+1 be an odd-dimensional totally umbilical submanifold of a Sasakian space form Men+1(c), 1 < m < n. If M2-+l is tangent to the contact vector 1; of Men+' (c) then M2'n+1 is a Sasakian space form immersed in Men+' (c) as a totally geodesic submanifold.

In contrast with Prop. 3.1 in [59], p. 260, a curvature-invariant (in the sense of K. Ogiue, [207], p. 389) submanifold M2m+1 of a Sasakian space form is always 1 and M2m+1 is tangent to 1;. Consequently, case b) in Theorem 1 of [60),

p. 225, has no analogue in contact geometry. In particular, cf. Theorem 17.1, there do not exist extrinsic spheres M2,11, m > 1, tangent to the structure vector l; of a Sasakian space-form. The following result holds (cf. S. Yamaguchi & H. Nemoto & N. Kawabata, [292], and I. Hazegawa, [127]):

Theorem 17.2 Let M2m+1 be a (2m + 1)-dimensional (1 < m < n) complete, simply connected, extrinsic sphere tangent to the contact vector l; of a Sasakian manifold M02n+1 If M2m+1 has a flat normal connection then M2m+1 is isometric to the standard sphere S2m+1(1/c) or radius 1/c, where c = ]]H]].

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Cf. B.Y. Chen, (54], p. 327, a k-dimensional Riemannian manifold Mk is sufficiently curved if for any x E Mk the maximal linear subspace V of the tangent space Tx(Mk) with the property R.,(u,v) = 0

for all u, v E V (here R is the curvature of Mk) has real algebraic dimension < k-2. In [79] it is shown that a submanifold M2m+1 bearing the hypothesis of Theorem 17.2 is subject to

R0(X,Y) = 0

(17.1)

for all tangent vector fields X, y on M2,+1, i.e. if M2`1 has codimension two (m = n - 1) then the ambient space M02n+1 is not sufficiently curved. Note that Sasakian space forms Men+1 (c) do not verify (17.1). We state (cf. [79], p. 162)

Theorem 17.3 Let MI be a real m-dimensional submanifold of the Sasakian manifold M02n+1. If MI is tangent to the contact vector of M02n+1 and has a parallel second fundamental form then M'n is a contact CR submanifold. In particular, any extrinsic sphere of M02n+1 is a contact CR submanifold.

Corollary 17.1 Let M' be a real m-dimensional submanifold of the Sasakian space form Men+1(c). Suppose Mm is tangent to and Vh = 0. Then either Mm is invariant (and thus totally geodesic) or MI is anti-invariant. The rest of this section is concerned with submanifolds of a g.H. manifold with a parallel second fundamental form. We report on a result of L. Ornea, [218]. The starting point is Theorem 17.3 above. We may state the following:

Theorem 17.4 Let M be a submanifold of a g.H. manifold Mon. If M is tangent to the Lee and anti-Lee vectors of Man and M has parallel second fundamental form then M is a CR submanifold. First, to illustrate the ideas leading to Theorem 17.4, we show how one may derive Theorem 17.4 in the context of a more restrictive class of g.H. manifolds compact and regular.

Let (Man, J, go) be a g.H. manifold. Set ((wo(( = 2c, c E (0, +oo). If Man is compact and regular (i.e. D' = RUo is regular, cf. Chapter 5) then N2n-1 = M02n/D1. Man is a principal S'-bundle over the Sasakian manifold The Sasakian structure of N2n-1 was described in Chapter 6 (cf. also I. Vaisman, (269]). Using standard lifting techniques associated with the S'bundle Man . N2n-1 and Theorem 17.3 one establishes

17.2. STABILITY

277

Corollary 17.2 Let M be a compact submanifold of a compact regular g.H. manifold M02'. If M is tangent to the Lee and anti-Lee vectors of M02 and has a parallel second fundamental form then M is a CR submanifold of M02. The proof of Corollary 17.2 is obtained simply by projecting M (via Mon -+

N2i-1) onto a submanifold N of N2n-1. Then, by Theorem 17.3, N must be a contact CR submanifold of N2n-1 The proof of Theorem 17.4 is more involved. One shows that FP = 0 and applies a result of K. Yano & M. Kon, [296], namely Theorem 3.1, p. 87. Note that Theorem 3.1 in [296] is stated for submanifolds of a Kahlerian manifold but is easily seen to hold for an arbitrary Hermitian ambient manifold. Cf. L. Ornea, [218], for details. Finally, we state (cf. [218], p. 8)

Theorem 17.5 Let Mon be a g.H.o manifold and M a submanifold of Mon with BJ- = 0 and R' = 0 and VJ-H = 0. If M has sectional curvature > 0 and II All = const., then M has a parallel second fundamental form.

17.2

Stability

Let M be a complex hypersurface of a g.H.o manifold with Bl = 0. A normal vector field l; E E(W) is recurrent if OX-LC = a(X)1;

for some 1-form a on M and any X E T(M). From now on, let M be compact. The normal variation v(C) induced by

(17.2)

vW Here So(e) = -Rico

is defined by

and 2n

Rico(X,Y) _ i=1

for any X, Y E T(Mon) and some local orthonormal frame {E;} on Mon. We recall that a minimal submanifold M is stable if v(e) > 0 for any E E(LY). Otherwise M is unstable. For a geometric interpretation, as well as a physical interpretation, of stability, see J.C. Nitsche, [199], vol. I, p. 97 and p. 115116. We have (cf. [133], p. 129):

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278

Theorem 17.6 Let M be a complex hypersurface of the g.H.o manifold Mpn. Assume that Bl = 0. If there is l E E(W) so that one of the following conditions is fulfilled i) V- = 0 and p!5 0 or ii) t: is recurrent with a = u and p < 0, then M is unstable. Here p = trace(Ric) and u = 11w II -1w. The proof of Theorem 17.6 is obtained by computing v(C) for a recurrent £ E E(fl. Cf. [133], p. 128-129 for details. The main observation is that one may effectively compute the Ricci form of M from the Gauss equation. Along the same line we have (cf. [87], p. 202-206)

Theorem 17.7 Let M3 be a compact minimal real hypersurface of nonnegative scalar curvature (p > 0) of the g.H.o manifold Mo (e.g. Mo = CH2). If M3 is tangent to the Lee field of Mp then M3 is stable. Indeed, a computation based on the Gauss equation shows that v(fN)=I {for

any compact orientable minimal real hypersurface M3 of the 1.c.K.0 man-

ifold M. Here N E E(I') is a unit normal field on M3 and f E C°O(M3) is arbitrary. Hence Theorem 17.7 follows from the more general observation: If Bl = 0 and Lo(N, N) < div(B) then M3 is stable.

17.3

f -Structures

Let Men+' be a real (2n + s)-dimensional differentiable manifold carrying an f-structure (cf. K. Yano, [2931), i.e. a (1,1)-tensor field f of rank 2n satisfying f3 + f = 0. Moreover f is said to have complemented frames if there

exist the 1-forms rla and the tangent vector fields fa on Men+', 1 < a < s, so that .f (ba) = 0 , '7°(W = bb 77a

0 f = 0, .f 2 = -I + rla ®ta

The f-structure is normal if [f, and it is metric if a Riemannian metric g on Me+' is prescribed in such a way that g(X, y) = g(f X, fY) + rla(X)rla(Y)

17.3. F-STRUCTURES

279

for any tangent vector fields X, Y on M2n+'. Throughout, if = rya. Let

F(X, Y) = g(X, fY) be the fundamental 2-form. Then (f l;a, ,a, g) is a K-structure if it is normal and F is closed (dF = 0). Also M2n+' is said to be a K-manifold. See H. Nakagawa, [197], and D.E. Blair, [27]. Let (M2n+s, f, £a, 77a, g) be a K-manifold. If

di1a = aaF

for some functions a0 E C°° (M2n+s) then (f, & ,a, g) is a S-structure (and M2n+' a S-manifold). If this is the case, then as E R, each !;a is Killing (with respect to g), and

201;0+aaf =0 , 1 1, S-manifold whose S-structure is regular, then there is a T'-principal bundle ir : M2n+' - Mon = M2n+s/M where Mpn is a Kahlerian manifold. Here T' is the s-torus. Also 7 = (,j,. , rls) is a connection 1-form in T' M2n+s . Mon. S. Goldberg, [110], has inaugurated a program of unifying the treatment of the cases s even and s odd, and has studied f-invariant submanifolds of codimension 2 of an S-manifold. Let W : M --+ M2n+' be an isometric immersion of a Riemannian manifold (M, g) in an S-manifold M2n+' with structure tensors (f, la, rya, go).

Then M is said to be a CR submanifold of M2n+' if it carries a smooth distribution D so that i) D is f-invariant (i.e. fx(Dy) C V, for any x E M) and ii) its orthogonal (with respect to g) complement Dl in T(M) is f-antiinvariant (i.e. fx(Dz) C E(W)x, for any x E M). Since f-structures generalize both almost complex (s = 0) structures and almost contact (s = 1) structures, clearly this notion of CR submanifold extends that of CR submanifold of an (almost) Hermitian manifold and that of contact CR submanifold of an

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280

a.ct.m. manifold (cf. [301]). A framed CR submanifold M of the S-manifold (M2' , f, a, 17a, go) is a CR submanifold which is tangent to the structure vectors Sa. See I. Mihai, [188], and L. Ornea, [220]. The case studied by

S. Goldberg, [110], corresponds to D1 = {0} (and codim(M) = 2). The assumption that M is tangent to each C. is rather natural, as motivated by the following:

Theorem 17.8 Let Mm be a real m-dimensional submanifold of the Smanifold Mo"+s , s > 1. If, for some 1 < a < s, the structure vector S. is normal to Mm and as j4 0 then Mm is anti-invariant (i.e. fx(TT(Mm)) C_ E(W)x, for any x E Mm) and 2m < 2n + s. Cf. [87], p. 195. One may further study the geometry of the second fundamental form of a submanifold in a S-manifold on the line indicated in [87] and [1]. To illustrate our ideas, we state

Theorem 17.9 Let Mm+' be a (m + s)-dimensional submanifold of the Smanifold Mo"+' with structure tensors (f, l a, ija, go). Assume that M'"+-' is tangent to the structure vectors Ca. Then 1) If Mm+' is invariant (fxTx(M'"+') C Tx(Mm+'), for any x E M'"+') and M02"+' = M2"+' (c) has constant f -sectional curvature c, then the following statements are equivalent: i) Mm+' is totally geodesic. ii) Mm+' is a S-manifold of constant f -sectional curvature c.

2) If M'"+' is totally umbilical then it is totally geodesic. If this is the case and as # 0 for some 1 < a < s then M'"+' is invariant. 3) If M'+' is invariant and has a parallel second fundamental form and as # 0 for some 1 < a < s then Mm+' is totally geodesic. 4) Assume that (f, ta, rla, go) is a C-structure, i.e. as = 0. If M'+' is anti-invariant then the following statements are equivalent: i) M'"+' has a flat normal connection.

ii) R4 = 0. Here Rf- is the L-component of the curvature R of (M'"+', g) with respect to

T(Mm+3) = G fl) M,

that is

R(X,Y)Z = Rl(X,Y)Z+ fa(X,Y,Z)ta for any X, Y, Z E T(M'"+'). Also ,C is the m-distribution on M'"+' given by the Pfaffian system rla = 0, 1 < a < s.

17.3. F-STRUCTURES

281

Cf. [871, p. 195-199. Let 7r1 : S2n+1 H2n+,,

CPn be the Hopf fibration and set

{(p1,.. .,p8) E S2n+1 x ... X S2n+1 : lr1V'1)

=

We define a principal toroidal bundle Try

as the pullback 0'

V'

: H2n+s , CPn

S2n+1 of the T'-bundle :S2n+1

7rl

_ ... = 7r1(pe)}

x... xS2n+1

via the diagonal map

A:CP"-, 0(x) _ (X, ... , x) , x E CPn. Here T' is the s-torus T' = S1 x x S' (s terms). The following diagram is commutative: H2n+s 7r3

41

S2n+1

x . . . X S2n+1

7r1x...x7r1

1

1

CPn

CPn X . . X CP"

where i is the inclusion. Let Q . S2n+l X ... X S2n+1

S2n+1

1 0) has been examined in Theorem 12.2. Let us prove Theorem 17.11. Suppose (OXP)Y = 0. Since Mm is generic

f = 0. Let us replace Y by PY in (12.9). We obtain

h(X,PY) = 2 [w(PY)PX - w(P2Y)X +g(X, P2Y)B - g(X, PY)PB - g(X, PY)tB1] (17.4)

for any X,Y E T(Mm). Now let Y E D. Since F(D) = 0 we derive, by (12.10)

tFVxY =

t h(X, PY) + 2 [w(Y)tFX

-g(X, Y)tFB + g(X, PY)tB']. At this point p2 + tF = -I and (17.4)-(17.5) lead to

tFOXY = 2 [w(PY)PX - g(X, PY)PB +g(X,Y)P2B - w(Y)P2X] and thus

FVXY=0

(17.5)

17.5. SECTIONAL CURVATURE

285

P[w(PY)X - w(Y)PX + g(X, Y)PB - g(X, PY)BI = 0. Suppose now that X E D. We obtain

w(Y)X + w(PY)PX = g(X, PY)PB - g(X,Y)P2B.

(17.6)

On the other hand, by (17.4) for X,Y E D, we have

t h(X, PY) = 2 [w(Y)X + w(PY)PX -g(X, Y)B - g(X, PY)PB - g(X, PY)tB1]. (17.7) Substitution from (17.6) into (17.7) leads to

h(X,Y) = -2[f2(X,Y)FB+g(X,Y)tB1]

(17.8)

for any X, Y E D. Since h, g are symmetric and S2 skew, it follows that

51(X,Y)FB = 0

for any X, Y E D. As M' is generic p = m - n. Then, either m = n and then Mm is anti-invariant, or m - n > 1. Let us show that actually the second case does not occur. Indeed, if m - n > 1 then dimR V > 2 and we may choose X = PPY E Dx , X # 0, at some x E M. Thus F1BT = 0, i.e. Bx E D. At this point we may replace Y by B in (17.6) (all calculations are carried out at x) so as to yield IIwI12X = 0, a contradiction, Q.E.D.

17.5

Sectional curvature

: G2(Mm) , Mm be the Grassmann bundle of all 2-planes tangent to Mm. Let Riem : G2(Mm) R be the (Riemannian) sectional curvature of (Mm, g). If (Mm, D) is a CR submanifold of the Hermitian manifold Mo" then a 2-plane po E G2(Mm) is anti-holomorphic if J(po) and po are orthogonal. If additionally po g D,(p0) then po is said to be D-anti-holomorphic. Next (cf. A. Bejancu, [15], p. 96) the D-anti-holomorphic sectional curLet -7r

vature of the CR submanifold (Mm, D) is the restriction of Riem to the D-anti-holomorphic planes of Mm. Moreover D1 is said to be D-parallel if VXY E D1 for any X E D and Y E D1. We may state the following (cf. [86], p. 5):

Theorem 17.13 Let (Mm, D) be a CR submanifold of the g.H.o manifold CH°(c). Let us assume that

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286

i) D1 is D-parallel; ii) there is a constant A > 0 so that 11h112 + 1IIWII2 < 2(c2 - A).

Then all D-anti-holomorphic sectional curvatures of M'n are > A. Also, cf. [84], p. 362, we may state Theorem 17.14 Let Mm be an m-dimensional submanifold of a g.H.o manifold CHn(c). If the scalar curvature p of Mm satisfies

p> (m-2)IIhII2+(m-2)(m-1)c2+2(m-1)A at a point x E Mm, for some A E R, then the sectional curvatures of M' are > A at the point x.

17.6

L. c. cosymplectic structures

As announced in Section 14.5, we wish to report briefly on the main results in [47]. First, we discuss odd-dimensional real Hopf manifolds. A similarity transformation of RI is given by

xpa'xr+b' where p > 0 and [a.7] E O(n). A manifold Mn is a local similarity manifold if it possesses a smooth atlas whose transition functions are similarity transformations (cf. [234]). Let 0 < A < 1 and let RHa = (Rn - {0})/0A be the real Hopf manifold. There is a diffeomorphism

f: RHa-+Sn-'xS1 V [x] E RHa : f ([x]) = (f.ex p

log

xl ))

Here [x] = 7r (x) , x E Rn - {0}, and 7r : R - {0} - RHa is the natural covering map. Note that RHa , n > 1, is a compact connected local similarity R2n+1- {0} manifold with the transition functions x" = \x*. Let us endow with the metric Go = (In12 + t2)-1 {6 dx' ® dx' + dt2}

17.6. L. C. COSYMPLECTIC STRUCTURES

287

where (x', t) , 1 < i < 2n, are the natural coordinates (cf. (4.4) in [234], p. 287). Since Go is invariant under any transformation

xt=Amx`

,

mEZ

it induces a globally defined metric go on RHan+1 We organize RHan+1 into a l.c. cosymplectic manifold as follows. Let a = log{Ix12 + t2}.

One may endow R2n+1 = R2n x R with a cosymplectic structure (cf. Z. Olszak, [210], p. 241). Namely, let g = b,j dx' ®dx1 + dt2

be the product metric on R2n+1 Let `P(X+fOt)=JX

where X is tangent to R2n and f E Coo ical complex structure of R2n " Cn. Also set

(R2n+1).

Here J denotes the canon-

17(X +f5) = f. Then (cp, £, 77, g), with t; = O/Ot, is a cosymplectic structure on R2n+1. Note

that a°t2t; , e-0'277 and a-Og are DA-invariant. Therefore RHan+1 inherits a l.c.c. structure (cpo, 6, r/o, go). Furthermore, by Prop. 3.5 in [234], p. 286, any orientable compact local similarity manifold of dimension m > 3 is a real homology real Hopf manifold, i.e. it has the Betti numbers bo = b1 = brn_1 =

By a theorem of D.E.Blair&S.I. Goldberg (cf. Theorem 2.4 in [29], p. 351), the Betti numbers of a compact cosymplectic manifold are nonzero. Combining the above statements one obtains in particular

Theorem 17.15 Any (2n + 1)-dimensional (n > 2) real Hopf manifold RHan+l has a natural structure of an l.c. cosymplectic manifold but admits no globally defined cosymplectic metrics. The Weyl connection of R.Han+1

is flat and its characteristic form w = dv is parallel with respect to the Levi-Civita connection of (RHan+l , go)

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288

Let us look now at the canonical foliation of an l.c.c. manifold. Let Mon be a real 2n. dimensional differentiable manifold. A (f, 9, u, v, A)-structure on Man consists of a (1, 1)-tensor field f, a R.iemannian metric g, two 1-forms u, v, and a smooth R-valued function A E C°O(Mon) subject to

f2=-I+u®U+v®V uo f =Av,vo f = -Au, fU= -AV, f V=AU u.(V)=v(U)=0,u(U)=v(V)=1-A2

g(fX, fY) = g(X,Y) - u(X)u(Y) - v(X)v(Y) where U = ut, V = vb (raising of indices is performed with respect to g), cf. [297], p. 386. Let (M2 n+1, cp, t;, g, go) be an l.c.c. manifold with the characteristic 1form w. Assume from now on that M2n+1 is strongly non-cosymplectic.

Then Men+1 admits a canonical foliation F whose leaves are the maximal connected integral manifolds of the Pfaffian equation w = 0. We may state (cf. [471)

Theorem 17.16 Each leaf of the canonical foliation F of a strongly noncosymplectic 1. c. c. manifold Men+1 has an induced (f , g, u, v, A)-structure whose 1-form v is closed. If the characteristic 1 form w of M2n+i is parallel then .F has totally geodesic leaves. If moreover the local cosymplectic metrics gi, i E I, of Men+1 are fiat then the leaves of F are Riemannian manifolds of constant sectional curvature. If additionally A'I2n+1 is normal, then each complete leaf of .F is holomorphically isometric to CPn(c2), c = IIwI]/2.

Let us discuss now regular l.c.c. manifolds. An l.c.c. manifold M2n+1 with the characteristic form w is normal if and only if w = W(077-

If this is the case, and M2n+1 is strongly non-cosymplectic, then . is regular if and only if B = wa is regular. We state (cf. [47])

Theorem 17.17 Let M2n+1 be a compact normal l.c.c. manifold. If the structure vector 6 is regular then i) M2n+1 is a principal Sl-bundle over the leaf space Mpn = M2n+1g; ii) the almost contact 1-form rl yields a flat connection 1-form on M2n+1; iii) the base manifold Mon has a natural structure of Kahlerian manifold.

Finally, we look at the real homology type of an l.c.c. manifold, and state the following (cf. [47]):

17.7. CHEN'S CLASS

289

Theorem 17.18 Let M2n+1 be a connected compact orientable (strongly non-cosymplectic) 1. c. c. manifold with a parallel characteristic 1 form w and a flat Weyl connection. Then the Betti numbers of M2n+1 are given by bo

(M2n+1) = b2n+1(M2n+1) = 1

b1(M2n+1)

=

b2n(M2n+1)

=1

bp(M2n+1) =0, 2 < p < 2n - 1 i.e. M2n+1 is a real homology real Hopf manifold.

The proof is similar to that of Theorem 6.5 in Chapter 6 (in that it exploits the explicit expression of the curvature tensor field of an l.c.c. manifold whose local cosymplectic metrics are flat (an expression similar to (6.5), giving the curvature of a g.H.o manifold) to compute Bochner's quadratic forms Fp(a)). The same technique has been used for studying the real homology type of a compact Riemannian manifold of quasi-constant curvature, cf. Theorem 2 in [89], p. 184.

17.7

Chen's class

In his note [53], BY. Chen has introduced a cohomology class c(M'n) E H2p(M'n, R) associated with a CR submanifold MI (of CR dimension p) of a Kahlerian manifold. He used c(Mm) to prove the following:

Theorem 17.19 Let (M',D) be a closed (i.e. compact, orientable) CR submanifold of a Kahlerian manifold, If b2j (Mm) = 0 for some j < dimc 1x,

x E Mm, then either the holomorphic distribution is not integrable or the anti-invariant distribution Dl is not minimal. We refer to c(Mm) (and its generalizations) as the Chen class of M'. Below, we briefly sketch its construction and indicate the way it may be used to establish Theorem 17.19. The arguments used to build c(Mm) carry over easily to the case of a CR submanifold in various ambient spaces. For instance, see B.Y. Chen & P. Piccinni, [64], for the Chen class of a CR submanifold of a 1.c.K. manifold. Cf. also G. Piti§, [230], for the Chen class of an anti-quaternion submanifold of a quaternion Kahler manifold. Let (Mm, D) be a CR submanifold of the Kahlerian manifold Mon. Let

{E1i...,Ep,JEI,...,JEp}

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290

be a local orthonormal frame of V and define the 1-forms wA, 1 < A < 2p, by setting WA(EB) = 6BA

where { EA } = {Ej, JET } and E3+p = JE3 , 1 < j < p. Then

w=w1A...Aw2p defines a 2p-form on Mm. This 2p-form is globally defined because D is orientable. Next, using the minimality of D, B.Y. Chen proves (cf. [53]) that w is closed thus giving rise to a De Rham cohomology class c(M'n) _ [w]. Moreover, if V is integrable and D-i is minimal then c(Mm) 34 0. Let Sl(X, Y) = g(X, PY) be the fundamental 2-form of MI. Since Mon is Kahlerian, dfl = 0, thus giving rise to a De Rham cohomology class [St) E H2(Mm; R). A straightforward calculation shows that [12]p = (-1)pp!c(Mm).

To prove Theorem 17.19, assume that b2.j(Mm) = 0 for some j < p. Then [0] = 0 (as [Si]j E H2j (M'"; R) = 0) and hence c(M'n) = 0. Thus either V is not integrable, or Dl is not minimal, Q.E.D.

Let us look now at the l.c.K. case. Let (Mm, V) be a CR submanifold of the strongly non-Kahler manifold Mgr. Assume that D1 C T(FFo), where .FO is the canonical foliation of M2n. Then (by Corollary 2.3 in [64], p. 291) V is minimal. Under these assumptions, let w = wl A A w2p be the volume

form of D, built as in the Kiihlerian case. The integrability of Dl and the minimality of D yield the following identities:

dw(Z1,Z2,X1,...,Y2p-1) = 0

(17.9)

dw(Z,,X1,...,X2p) =0

(17.10)

for any Z1, Z2 E Dl and any X1, dw = 0. We may state

, X2p E D. Then (15.9)-(15.10) yield

Theorem 17.20 (Cf. [64]) Let Mpn be a strongly non-Kahler l.c.K. manifold. Let (Mm, D) be a CR submanifold of Mon with Dl C T(.Fo). Then the 2p-form w is closed and defines a De Rham cohomology class c(Mm) _ [w] E H2p(Mm, R). If Dl is minimal and V is integrable then i) c(Mm) 54 0; ii) if in addition Mm is contained in a leaf of Fo then H2k(Mm, R) 36 0 for

any0 0, 2 < p < n. Then CR submanifold of R2i-1(-3). Hence S' X R2("-p)-l x T(rl, , rp) is a R2("-p)-l

298

CHAPTER 17. MISCELLANEA

generic CR submanifold of the Vaisman manifold S' X R2n-1(-3). It is tangent to the Lee field of S' x R2r-1(-3) and submersed over the Vaisman R2(n-p)-1 manifold S1 x S2n-1, it follows that S2n-1 (with 2) As CHe is isometric to S'(1/n) X the natural Riemannian structure) is isometrically immersed in CHe`s (with the Boothby-Wang metric). As such, S2n-1 is a generic CR submanifold of CHe2, normal to the Lee field of CH,, and submersed over CPs-1 with the Fubini-Study metric.

Appendix A

Boothby-Wang fibrations In this appendix we briefly present the necessary background of contact geometry and, especially, the Boothby-Wang fibration. A good reference for this topic is the book [25]. Let N be a (2n + 1)-dimensional smooth manifold. A contact form on N is a one-form 77 satisfying (di)n A rl # 0,

i.e. the 2n-dimensional distribution V generated by the vector fields which annihilate 77 is `as far from being integrable as possible'. We say that 77 endows N with a contact structure. We note that the maximal dimension of an integral manifold of D is n and it can be effectively attained. Any smooth hypersurface immersed in R2n+2 such that none of its tangent spaces contains the origin of R2n+2 has a contact structure. In particular, the odd-dimensional spheres have a contact structure which, being invariant under reflections through the origin, induces a contact structure on RP2n+1 Obviously the contact form induces an orientation on N and also on D. Hence N admits a global non-vanishing vector field, usually denoted , such that 77(1;) = 1. If is regular on N, then the contact structure (and, by abuse, N itself) is called regular. The above examples are regular; instead, no odd-dimensional torus can carry a regular contact structure. In the regular case, if N is also compact, its geometry is closely related to that of the quotient manifold B = N/ of all orbits of C. More precisely,

Theorem A.1 (cf. [37]) i) Let (N, rl) be a compact, regular, contact manifold. Then N is a principal circle bundle over B and 77 is a connection form in this bundle. The curvature

300

APPENDIX A. BOOTHBY-WANG FIBRATIONS

form S2 of 77 defines a symplectic form on B and determines an integral cocycle.

ii) Conversely, let B be a compact 2n-dimensional symplectic manifold whose symplectic form w determines an integral cohomology class. Then there exists a principal circle bundle rr : N - B and a connection form rl in it which is a contact form on N. Moreover, the associated vector field , of rl generates the right translations of the structural group S1 of this bundle.

The usual Hopf fibration S2n+1 -' CP" is but one example. Let us sketch the proof of Theorem A. 1. First, a gauge transformation may be performed on rl such that the new obtained generate the circle group S'. Now we describe the bundle structure. By regularity, we can choose coordinate neighbourhoods Ui with coordinates x1, ... , x2n+1 such that x1 = const., , x2s = const. represent the integral curves of C. Then {rr(Ui)} is an open cov-

ering of B with coordinates identified with {x', ,x2n}. We can define local cross-sections on {rr(Ui) } by setting si(xl'...'X 2n) = (xl,... , x2n, c) for some constant c. The maps 4)i : rr(Ui) x S1 -+ M, 4)i(p,t) = 4tsi(p), 4t being elements of the one-parameter group of diffeomorphisms generated by are the coordinate functions of the bundle. To show that rl is a connection form in this bundle, note first that Zt77 = 0 and LCdr7 = 0, hence 77 and dr7 are invariant under the action of S. Then, the Lie algebra of S1 is identified with R with basis A = d/dt and we must show that 77®A (denoted, for simplicity, by 77) is a connection form. This is implied by the two equations (i) rl(A*) = A, where A* is the fundamental vector field corresponding to A and (ii) by equivariance under right translations. (ii) is clear because the right translation by t E S' is just Ot. For (i) we need only

observe that A* = . Let S2 be the curvature form of 77. As St is Abelian [71, r7] = 0, hence the structure equation reads dr7 = ft. But d77 is horizontal, invariant under

the action of S', and thus basic. Therefore there exists l on B such that rr*S2 = A. Then rr*dfl = drr*S2 = d2i7 = 0 yielding dSl = 0. Finally rr*S2" _ (rr*SZ)" = (dr7)" # 0 so that S2 is a symplectic structure on B. Finally, a result in [160] proves that [S2] E H2(B, Z).

To prove the converse, recall that the principal S1 bundles over B are parametrized by H2(B, Z) (see e.g. [160]). So we construct the circle bundle corresponding to [S2] and let 77' be a connection form in it. Let 1' be the twoform on B with dr7' = rr*S2'. But [S2] = [S2'] thus S2 = S2' + dip, r' E A1(B). We let 77 = r7' + it*Vi. It can be shown that 77 is invariant to right translations and d77 = rr*fl. Moreover, if 6 is vertical such that rf (t;) = 1, then also 77(l;) = 1. To prove that 77 is indeed a contact form on N, we fix an arbitrary

APPENDIX A. BOOTHBY-WANG FIBRATIONS

point x E N and choose vl, Then

301

, v2n independent horizontal vectors in T.N.

(dg)" A 77(v1,...,v2n,S)

T0

and this ends the proof. Suppose now a Riemannian metric was choosen on N such that _ 77a (this is always possible). Then (N,1), g) is a metric contact manifold If C is Killing, denoting with V the Levi-Civita connection of g, we can define a smooth section End(TN), of rank 2n, by the formula cp = VC. The following relations are easily verified:

P2 =-I+rl(&£,

0

(A.1)

9(cPY, VZ) = 9(I', Z) - il(Y)rl(Z).

(A.2)

(DYV)Z = 17(Z)Y - 9(Y, Z)

(A.3)

If, moreover,

on N is satisfied, then (N, rl, g) is a Sasakian manifold. The above condition has the following geometric meaning. On the product N x R one naturally defines an almost complex structure by the formula J(Y, d/dt)= (cpY 17(Y)d/dt). Then the product metric is Kiihlerian with respect to this J if the structure on N is Sasakian. The usual contact structure of the odd-dimensional sphere is Sasakian. However, on S5 there exists also a non-Sasakian contact structure. The first Betti number of a compact Sasakian manifold is zero or even, thus the 3-dimensional torus cannot admit a Sasakian structure. Note that this also implies that compact g.H. manifolds with regular Lee fields have an odd first Betti number. Other examples can be produced using the Boothby-Wang fibration 7r : (B, Q). Namely, suppose the symplectic structure f of B derives (N, 17') from a Ki hlerian structure (J, h) (i.e. fl is the fundamental 2-form of h). Then we can define a new contact form n on N with d17 = n*St. The metric h can now be lifted by letting

g=7r*h+r/®q. Moreover, the associated endomorphism cp is given by WY = (J7r*Y)* where

Y*is the horizontal lift of Y. It can be verified that the structure thus obtained is Sasakian.

302

APPENDIX A. BOOTHBY-WANG FIBRATIONS

Note that, in general, a symplectic structure need not come from a Kahlerian one. But we can always find an almost Kahlerian structure inducing it. The contact structure induced as above on the total space of the Boothby-Wang fibration is called K-contact. The converse procedure is also possible. If the total space of the BoothbyWang fibration is Sasakian, one can project the metric on the base such that Tr becomes a Riemannian submersion because is Killing. Moreover from equation (A.3) we derive 0, hence W is projectable too, yielding an almost complex structure, compatible with the projected metric by (A.2). The same equation (A.3) projected implies that the Hermitian structure thus obtained is Kahlerian. Summing up and taking into account Theorem A.1 we obtain the following result of Y. Hatakeyama, [128], and A. Morimoto, [193]:

Theorem A.2 The total space of the Boothby- Wang fibration is Sasakian if and only if the base space is a Hodge manifold.

Appendix B

Riemannian submersions Let (M, g) and (M', g') be two Riemannian manifolds and 7r : M - M' a surjective C°° submersion. If x E M then V. = Ker(dx7r) is the vertical space at x and its orthogonal complement with respect to gx, ?-lx = V2 C T1(M) is the horizontal space at x. We call 7r a Riemannian submersion if dxir : Nx -' T,r(x)(M') is a linear isometry of (fx,gx) onto (Trlxl(M'),gAixl) for any x E M. A tangent vector field X E X (M) is vertical (respectively horizontal) if Xx E Vx (respectively Xx E 7-lx) for any x E M. Next X is a basic vector field if it is horizontal and 7r-related to some tangent vector field X' E X(M'), i.e. (dx7r)Xx = X'.(.) for any x E M. Let h : T(M) 71 and v : T(M) -+ V denote the natural projections associated with the direct sum decomposition T (M) = 7-l ® V. Let V be the Levi-Civita connection of (M, g). We define a (1, 2)-tensor field T on M by setting

TXY = hV xvY + for any X, Y E X (M). The pullback of T to each fibre it-1 (p) is the second fundamental form of 7r-1 (p) in (M, g). Moreover, at each point x E M, (TT)x is a skew-symmetric linear operator of Tx(M) into itself and it reverses the horizontal and vertical spaces. Also T is vertical, in the sense that TX = TVX for any X E X (M), and symmetric on vertical vectors, i.e. TXY = TyX for all X, Y E V. The last fact, which is well known for the second fundamental form of a submanifold, follows easily from the integrability of V. We may state the following elementary but useful result (cf. Lemma 1 in [214), p 460):

Proposition B.1 Let X, Y E X(M) be two basic vector fields which are ir-related to X', Y' E X (]W), respectively. Then

304

APPENDIX B. RIEMANNIAN SUBMERSIONS

1) g(X, Y) = g'(X', Y') o 7r; 2) h[X, Y] is a basic vector field 7r-related to [X', X'], 3) hVxY is a basic vector field 7r-related to VX,Y'.

Here V' denotes the Levi-Civita connection of (M', g'). The integrability tensor A of 7r : M -' M' is defined by AxY = hVhXVY + VVhxhY

for any X, Y E X(M). At each x E M, (Ax)x is a skew-symmetric linear operator of Tx(M) in itself and it reverses the horizontal and vertical spaces. Moreover A is horizontal, in the sense that Ax = AhX for any X E X (M), and skew-symmetric on horizontal vectors, i.e. AXY = -AyX for all X, Y E N. By a result of B. O'Neill, [214], if X,Y are horizontal vector fields on M then

AxY =

1 V[X, Y],

hence N is involutive if and only if A = 0. We recall the following fundamental equations of a Riemannian submersion, cf. [214]. If X, Y, Z E V and x E M then we define (R(X,Y)Z) to be the curvature of the fibre 7r-1(p) x at x, where p = 7r(x). Then

g(R(X,Y)Z,W) = g(R(X,Y)Z,W) -g(TxZ, TyW) + g(TyZ, TxW) g(R(X, Y)Z, U) = g((VyT)(X, Z), U) - g((VxT) (Y, Z), U)

(B.1) (B.2)

for any X, Y, Z, W E V and any U E R. The restrictions of (B.1)-(B.2) to a 7r_1(Y) are the Gauss and Codazzi equations of 7r-1(p) in M, cf. e.g. fibre

[52]. Let R' be the curvature tensor field of (M',g') and R' its horizontal lift to M, i.e. R'(X1, X2, X3, X4) = g'(R'(XI', X2)X3, X4) o 7r

for any Xi E N where Xi'n(x) _ (dxir)Xi,x for any x E M. Then (cf. Theorem 2 in [214], p. 463)

g(R(X,Y)Z,W) = R`(X,Y,Z,W)-2g(AxY,AzW) +g(AyZ, AxW) + g(AzX, AyW)

(B.3)

g(R(X,Y)Z,V) = g((VzA)(X,Y),V)+g(AxY,TvZ) -g(AyZ,TvX) - g(AzX,TvY)

(B.4)

APPENDIX B. RIEMANNIAN SUBMERSIONS

305

for any X, Y, Z, W E l and any V E V. Finally, the last fundamental equation (cf. Theorem 3 in [214], p. 465) is

g(R(V,W)X,Y) = g((VxT)(V,W),Y)+g((VvA)(X,Y),W) -g(TvX,TwY) +g(AxV,AyW)

(B.5)

for any X, Y E 7-l and any V, W E V. Let X E M and p = 7r(x). If V, W E V, then k(V AW) will denote the sectional curvature of the 2-plane tangent to 7r-1(p) spanned by {V, W). Also, let k, k' be the sectional curvatures of M, M' respectively. As an application

of the fundamental equations (B.1)-(B.5) one may establish (cf. Corollary 1 in [214], p. 465) the following relations among the sectional curvatures of the total space, base manifold, and fibres of a given Riemannian submersion: k(V A W) = k(V A W) - g(TyV'Tww)

2ITvWII2

(B.6)

IIVAWIl

k(X AV)IIXII2IIVI12 =g((VxT)(V,V),X)+IIAxVII2- IITvXII2

(B.7)

k(X A Y) = k'(X'n Y') - 3IIAxYII2 y112

(B.8)

for any X, Y E 7-Lx and any V, W E V, where X' = (dx7r)X , Y' = (d.,7r) Y'

and x E M. The third equation (B.8) often gives a quite efficient way of computing the sectional curvature of (M', g'). The geometric meaning of (B.8) is that Riemannian submersions are curvature increasing on horizontal tangent 2-planes. To illustrate the usefulness of (B.8) we look at the following examples. 1) Let it : S2s+1 -+ CP"' be the Hopf fibration. When S2n+1 and CPs are endowed with the natural Riemannian metric (induced from the Euclidean structure of R2n+2) and the Fubini-Study metric, it becomes a Riemannian Then submersion. Let e = x=8/8x` be the outward unit normal on S2s+1.

J is tangent to S2s+1 and spans V = Ker(dir). The fibres of it are great Therefore, the (1,2)-tensor T circles in S2s+1, hence geodesics of vanishes. Also, if X, Y E ?f (i.e. X, Y are orthogonal to JC) then S2s+1.

AxY = g(X, JY)Je , AxJC = JX where J is the complex structure on C"+1. As it is the restriction to S2s+1 of a holomorphic map, when X is a basic vector field on S2s+1, so does JX. Using (B.8) it follows that the sectional curvature of the 2-plane tangent to CP" spanned by {X, Y} is 1 + 3g(X, JY)2II X A YII-2.

306

APPENDIX B. RIEMANNIAN SUBMERSIONS

2) Let G be a Lie group endowed with a bi-invariant Riemannian metric. Let H C G be a closed subgroup. Then the homogeneous space G/H has a naturally induced Riemannian structure and it : G -, G/H is a Riemannian submersion with totally geodesic fibres (hence T = 0). Again (B.8) may

be used to compute the sectional curvature of G/H. Let g, h be the Lie algebras of G, H, respectively. If X, Y are left invariant vector fields on G, i.e. X, Y E h1 C g, then AXY = (1/2)v[X, Y] E h. Also, it is known that the sectional curvature of a 2-plane tangent to G spanned by X, Y is k(X A Y) = (1/4)IIX A YII-211[X,Y]112. Then, by (B.8),

k'(X'AY')IIXAY112 == 4II[X,Y]II2+3IIv[X,Y]II2 where X' = (d7r)X and Y' = (da)Y.

The tensor fields T and A may be shown to control the submersion a very much like the second fundamental form controls an immersion. Namely, one has (cf. Theorem 4 in [214], p. 468)

Theorem B.1 Let it and fr be Riemannian submersions of a connected Riemannian manifold M onto another Riemannian manifold M'. If it and Ir have the same tensor fields T and A and if dit = d fr for some x E M, then 7r=Fr.

The simplest example of a Riemannian submersion is the projection of a Riemannian product manifold on one of its factors. A Riemannian submerM' is said to be trivial if it differs from such a projection sion it : M only by some isometry of M. A necessary condition for it : M - M' to be trivial is that both T, A vanish. However, this is not sufficient as shown by the example of a flat Mobius band projecting onto its central circle. Let 7r : M - M' be a Riemannian submersion with M complete. Let M' be a geodesic segment in M' and Cx : [a, b] M its unique C : [a, b] horizontal lift issuing at x E 7r-'(C(a)) (i.e. it o Cs = C and dC2/dt E 11). Let us fix a base point p E M'. The submersion group of it is the group G(7r) consisting of all C°° diffeomorphisms c : 7r-'(p) - 7r-(p) given by cbc (x) = Cy (b) as C ranges over all geodesic loops at p. Then (cf. Theorem 5 in [214], p. 469):

Theorem B.2 Let Tr : M -. M' be a Riemannian submersion of a complete Riemannian manifold M onto M'. Then it is trivial if and only if T = 0 and G(7r) = 0. The proof of Theorem B.2 relies on some results due to R. Hermann, [129].

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Locally Conformal Kahler Geometry

This is a unique monograph covering topics in complex geometry, an area of mathematical growth in recent years. The latest topics are addressed systematically, bringing us to the cutting edge in the mathematics of locally conformal Kahler (I.c.K.) manifold theory. Locally Confionnal hiihrlerGeomety is a differential geometric study of l.c.K. manifolds (i.e., manifolds carrying some l.c.K. metric) and their submanifolds. While the latest results on Vaisman's conjectures, spectral geometry of generalized Hopf manifolds. harmonic and holomorphic forms of l.c.K. manifolds, and pseudoharmonic maps or Hermitian surfaces are reviewed throughout the mathematics literature, here they are presented in a systematic manner, and many specific examples (e.g., complex Hopf manifolds, Inoue surfaces, and Thurston's manifolds) are discussed from this wider perspective.

The accent in submanifold theory is on the modern results on CauchyRiemann (CR) submanifolds of I.c.K. manifolds, for example, the classification of extrinsic spheres of a generalized Hopf manifold, the classification of compact minimal CR submanifolds, generically embedded in a complex Hopf manifold with the Boothby metric. This monograph explores the interrelation between l.c.K. metrics and other intensely studied geometric objects. such as Sasakian metrics, cosymplectic and locally conformal cosymplectic structures. f-structures, Chen's class, geodesic symmetries, and Kobayashi's submersed CR submanifolds.

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ISBN 0-8176-4020-7

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