The geometry of curvature homogeneous pseudo-Riemannian manifolds

389 Pages·2007·1.677 MB·English

by  Peter B. Gilkey

The Geometry of
Curvature Homogeneous
Pseudo-Riemannian Manifolds
ICP Advanced Texts in Mathematics ISSN 1753-657X
Series Editor: Dennis Barden (Univ. of Cambridge, UK)
Published
Vol. 1 Recent Progress in Conformal Geometry
by Abbas Bahri & Yong Zhong Xu
Vol. 2 The Geometry of Curvature Homogeneous
Pseudo-Riemannian Manifolds
by Peter B. Gilkey
EH - The Geometry of Curvature.pmd 2 2/22/2007, 3:55 PM
ICP Advanced Texts in Mathematics – Vol. 2
The Geometry of
Curvature Homogeneous
Pseudo-Riemannian Manifolds
Peter B. Gilkey
University of Oregon, USA
Imperial College Press
ICP
Published by
Imperial College Press
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Covent Garden
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
THE GEOMETRY OF CURVATURE HOMOGENEOUS
PSEUDO-RIEMANNIAN MANIFOLDS
ICP Advanced Texts in Mathematics — Vol. 2
Copyright © 2007 by Imperial College Press
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ISBN-13978-1-86094-785-8
ISBN-101-86094-785-9
Printed in Singapore.
EH - The Geometry of Curvature.pmd 1 2/22/2007, 3:55 PM
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Preface
This book arose out of a desire to investigate the relationship between
certain algebraic properties of the curvature tensor and the underlying ge-
ometry of a pseudo-Riemannian manifold.
In Chapter 1, we discuss the geometry of the Riemannian curvature
tensor. Basic geometrical notions are presented in Section 1.2. In Section
1.3, a passage to the algebraic context is given by introducing algebraic
curvature tensorswhich arealgebraicobjectswith the samesymmetries as
thatoftheRiemanncurvaturetensor. Onesaysthatapseudo-Riemannian
manifold is curvature homogeneous if the curvature tensor \looksthe same
at each point". This notion is made precise in Section 1.4. Section 1.5
presentssomeresultsfromlinearalgebraandSection1.6providesconcepts
fromdi(cid:11)erentialgeometrythatwillbeneededsubsequently. InSection1.7,
the geometry of the Jacobi operator is discussed and in Section 1.8, corre-
spondingresultsforthe curvatureoperatoraregiven. Chapter1concludes
in Section 1.9 with some general facts concerning the spectral geometry of
the curvature tensor.
Chapter 2 deals with curvature homogeneous generalized plane wave
manifolds. This is a class of manifolds that are geodesically complete,
whose exponential map is surjective, and which have a number of other
properties. In Section 2.2, we present the main geometrical results con-
cerning this class of manifolds. The remainder of Chapter 2 deals with
speci(cid:12)c examples. Sections 2.3, 2.4 and 2.5 deal with manifolds of signa-
ture (2;2), (2;4), and (p;p), respectively. Section 2.6 treats generalized
planewavemanifoldswith (cid:13)atfactors, Section2.7discussesNik(cid:20)cevi(cid:19)cman-
ifolds,andSection2.8presentsDunn manifolds. Sections2.9and2.10deal
with k-curvature homogeneous manifolds.
Chapter3discussesexampleswhicharenotgeneralizedplanewaveman-
v
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vi The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds
ifolds. Section 3.2 discusses Lorentz manifolds, Section 3.3 treats Walker
manifolds of signature (2;2), Section 3.4 deals with questions of geodesic
completenessandRicciblowup,andSection3.5presentsFiedlermanifolds.
Chapter4ismorealgebraicinnature. InSection4.2,wepresentvarious
topological results. In Section 4.3, we use the Nash embedding theorem to
providegeneratorsforthespaceofalgebraiccurvaturetensorsandalgebraic
covariant derivative tensors. Sections 4.4 and 4.5 treat Jordan Osserman
algebraiccurvature tensorsandSzabo(cid:19)covariantderivativealgebraiccurva-
ture tensors, respectively. In Section 4.6, we study questions in conformal
geometry. Section 4.7 deals with Stanilov models. Section 4.8 treats com-
plex geometry.
Chapter 5 contains a discussion of complex models which are both Os-
serman and complex Osserman; the classi(cid:12)cation is complete except in a
few exceptional dimensions and ranks. Chapter 6 contains an introduc-
tion to Stanilov{Tsankov theory; this is a discussion of Jacobi Tsankov
manifolds, skew Tsankov manifolds, Stanilov{Videv manifolds, and Jacobi
Videvmanifolds. Chapters5and6arejointworkwithM.Brozos-Va(cid:19)zquez.
Eachchapter is divided into sections; the (cid:12)rst section of a chapter pro-
vides an outline to the subsequent material in the chapter. Many sections
are further divided into subsections. Theorems, lemmas, corollaries, and
so forth are labeled by section. Equations which are cited are labeled by
section; equations which are not cited are not labeled.
Much of this book reports on previous joint workwith variousauthors.
It is an honor and a privilege to acknowledge the contribution made by
these colleagues: N. Bla(cid:20)zi(cid:19)c,N. Bokan, M. Brozos-Va(cid:19)zquez,J.D(cid:19)(cid:16)az-Ramos,
C. Dunn, B. Fiedler, E. Garc(cid:19)(cid:16)a-R(cid:19)(cid:16)o, R. Ivanova, J. V. Leahy, Z. Raki(cid:19)c, H.
Sadofsky, U. Semmelman, U. Simon, G. Stanilov, I. Stavrov, A. Swann, L.
Vanhecke, V. Videv, and T. Zhang. In addition to pleasant professional
collaborationsthey have also enriched the personal life of the author.
TheauthorexpresseshisappreciationtotheMaxPlanckInstituteinthe
Mathematical Sciences (Leipzig, Germany) where most of the writing and
research took place. The author acknowledges with pleasure the support
and encouragementof LorraineDavis, Susana Lo(cid:19)pez-Ornat, Gwen Steigel-
man,andArnieZweig;withouttheseindividuals, the bookwouldnothave
been written. The book is dedicated to my father and mother.
P. B. Gilkey
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Contents
Preface v
1. The Geometry of the Riemann Curvature Tensor 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic Geometrical Notions . . . . . . . . . . . . . . . . . . . 4
1.2.1 Vector spaces with symmetric inner products . . . . 4
1.2.2 Vector bundles, connections, and curvature . . . . . 6
1.2.3 Holonomy and parallel translation . . . . . . . . . . 10
1.2.4 A(cid:14)ne manifolds, geodesics, and completeness . . . . 11
1.2.5 Pseudo-Riemannian manifolds . . . . . . . . . . . . . 12
1.2.6 Scalar Weyl invariants . . . . . . . . . . . . . . . . . 15
1.3 Algebraic Curvature Tensors and Homogeneity . . . . . . . 16
1.3.1 Algebraic curvature tensors . . . . . . . . . . . . . . 17
1.3.2 Canonical curvature tensors . . . . . . . . . . . . . . 21
1.3.3 The Weyl conformal curvature tensor . . . . . . . . . 23
1.3.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.5 Various notions of homogeneity . . . . . . . . . . . . 26
1.3.6 Killing vector (cid:12)elds . . . . . . . . . . . . . . . . . . . 27
1.3.7 Nilpotent curvature . . . . . . . . . . . . . . . . . . . 28
1.4 Curvature Homogeneity { a Brief Literature Survey. . . . . 28
1.4.1 Scalar Weyl invariants in the Riemannian setting . . 28
1.4.2 Relating curvature homogeneity and homogeneity . . 29
1.4.3 Manifolds modeled on symmetric spaces . . . . . . . 30
1.4.4 Historical survey . . . . . . . . . . . . . . . . . . . . 31
1.5 Results from Linear Algebra . . . . . . . . . . . . . . . . . . 32
1.5.1 Symmetric and anti-symmetric operators . . . . . . . 32
vii
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viii The Geometry ofCurvature Homogeneous Pseudo-Riemannian Manifolds
1.5.2 The spectrum of an operator . . . . . . . . . . . . . 32
1.5.3 Jordan normal form . . . . . . . . . . . . . . . . . . 33
1.5.4 Self-adjoint maps in the higher signature setting. . . 34
1.5.5 Technical results concerning di(cid:11)erential equations . . 35
1.6 Results from Di(cid:11)erential Geometry . . . . . . . . . . . . . . 38
1.6.1 Principle bundles . . . . . . . . . . . . . . . . . . . . 39
1.6.2 Geometric realizability . . . . . . . . . . . . . . . . . 39
1.6.3 The canonical algebraiccurvature tensors . . . . . . 41
1.6.4 Complex geometry . . . . . . . . . . . . . . . . . . . 47
1.6.5 Rank 1-symmetric spaces . . . . . . . . . . . . . . . 51
1.6.6 Conformal complex space forms . . . . . . . . . . . . 53
1.6.7 Ka(cid:127)hler geometry . . . . . . . . . . . . . . . . . . . . 54
1.7 The Geometry of the Jacobi Operator . . . . . . . . . . . . 54
1.7.1 The Jacobi operator . . . . . . . . . . . . . . . . . . 55
1.7.2 The higher order Jacobi operator . . . . . . . . . . . 57
1.7.3 The conformal Jacobi operator . . . . . . . . . . . . 59
1.7.4 The complex Jacobi operator . . . . . . . . . . . . . 60
1.8 The Geometry of the Curvature Operator . . . . . . . . . . 62
1.8.1 The skew-symmetric curvature operator . . . . . . . 62
1.8.2 The conformal skew-symmetric curvature operator . 65
1.8.3 The Stanilov operator . . . . . . . . . . . . . . . . . 66
1.8.4 The complex skew-symmetric curvature operator . . 66
1.8.5 The Szabo(cid:19) operator . . . . . . . . . . . . . . . . . . . 68
1.9 Spectral Geometry of the Curvature Tensor . . . . . . . . . 69
1.9.1 Analytic continuation. . . . . . . . . . . . . . . . . . 70
1.9.2 Duality. . . . . . . . . . . . . . . . . . . . . . . . . . 72
1.9.3 Bounded spectrum . . . . . . . . . . . . . . . . . . . 75
1.9.4 The Jacobi operator . . . . . . . . . . . . . . . . . . 78
1.9.5 The higher order Jacobi operator . . . . . . . . . . . 81
1.9.6 The conformal and complex Jacobi operators . . . . 82
1.9.7 The Stanilov and the Szabo(cid:19) operators . . . . . . . . 83
1.9.8 The skew-symmetric curvature operator . . . . . . . 84
1.9.9 The conformal skew-symmetric curvature operator . 86
2. Curvature Homogeneous Generalized Plane Wave Manifolds 87
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2 Generalized Plane Wave Manifolds . . . . . . . . . . . . . . 90
2.2.1 The geodesic structure . . . . . . . . . . . . . . . . . 92
2.2.2 The curvature tensor . . . . . . . . . . . . . . . . . . 93
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Contents ix
2.2.3 The geometry of the curvature tensor . . . . . . . . . 94
2.2.4 Local scalar invariants . . . . . . . . . . . . . . . . . 94
2.2.5 Parallelvector (cid:12)elds and holonomy . . . . . . . . . . 96
2.2.6 Jacobi vector (cid:12)elds . . . . . . . . . . . . . . . . . . . 96
2.2.7 Isometries . . . . . . . . . . . . . . . . . . . . . . . . 97
2.2.8 Symmetric spaces . . . . . . . . . . . . . . . . . . . . 99
2.3 Manifolds of Signature (2;2). . . . . . . . . . . . . . . . . . 101
2.3.1 Immersions as hypersurfaces in (cid:13)at space. . . . . . . 103
2.3.2 Spectral properties of the curvature tensor . . . . . . 105
2.3.3 A complete system of invariants . . . . . . . . . . . . 107
2.3.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . 109
2.3.5 Estimating k if min(p;q)=2 . . . . . . . . . . . . 114
p;q
2.4 Manifolds of Signature (2;4). . . . . . . . . . . . . . . . . . 115
2.5 Plane Wave Hypersurfaces of Neutral Signature (p;p) . . . 119
2.5.1 Spectral properties of the curvature tensor . . . . . . 123
2.5.2 Curvature homogeneity. . . . . . . . . . . . . . . . . 128
2.6 Plane Wave Manifolds with Flat Factors . . . . . . . . . . . 130
2.7 Nik(cid:20)cevi(cid:19)c Manifolds . . . . . . . . . . . . . . . . . . . . . . . 135
2.7.1 The curvature tensor . . . . . . . . . . . . . . . . . . 137
2.7.2 Curvature homogeneity. . . . . . . . . . . . . . . . . 139
2.7.3 Local isometry invariants. . . . . . . . . . . . . . . . 141
2.7.4 The spectral geometry of the curvature tensor . . . . 145
2.8 Dunn Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . 149
2.8.1 Models and the structure groups . . . . . . . . . . . 151
2.8.2 Invariants which are not of Weyl type . . . . . . . . 155
2.9 k-Curvature Homogeneous Manifolds I . . . . . . . . . . . . 156
2.9.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . 159
2.9.2 A(cid:14)ne invariants. . . . . . . . . . . . . . . . . . . . . 162
2.9.3 Changing the signature . . . . . . . . . . . . . . . . . 164
2.9.4 Indecomposability. . . . . . . . . . . . . . . . . . . . 165
2.10k-Curvature Homogeneous Manifolds II . . . . . . . . . . . 166
2.10.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . 168
2.10.2 Isometry groups . . . . . . . . . . . . . . . . . . . . . 171
3. Other Pseudo-Riemannian Manifolds 181
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.2 Lorentz Manifolds . . . . . . . . . . . . . . . . . . . . . . . 182
3.2.1 Geodesics and curvature . . . . . . . . . . . . . . . . 185
3.2.2 Ricci blowup . . . . . . . . . . . . . . . . . . . . . . 187