Riemannian Geometry of Contact and Symplectic Manifolds (eBook)

Preface to Second Edition 7
Preface to the First Edition 8
Contents 10
Symplectic Manifolds 15
1.1 Definitions and examples 15
1.2 Lagrangian submanifolds 20
1.3 The Darboux–Weinstein theorems 23
1.4 Symplectomorphisms 25
Principal S1-bundles 28
2.1 The set of principal S1-bundles as a group 28
2.2 Connections on a principal bundle 32
Contact Manifolds 35
3.1 Definitions 35
3.2 Examples 39
3.3 The Boothby–Wang fibration 48
3.4 The Weinstein conjecture 50
Associated Metrics 53
4.1 Almost complex and almost contact structures 53
4.2 Polarization and associated metrics 57
4.3 Polarization of metrics as a projection 61
4.4 Action of symplectic and contact transformations 69
4.5 Examples of almost contact metric manifolds 72
Integral Submanifolds and Contact Transformations 80
5.1 Integral submanifolds 80
5.2 Contact transformations 82
5.3 Examples of integral submanifolds 85
Sasakian and Cosymplectic Manifolds 90
6.1 Normal almost contact structures 90
6.2 The tensor field 94
6.3 Definition of a Sasakian manifold 97
6.4 CR-manifolds 100
6.5 Cosymplectic manifolds and remarks on the Sasakian definition 106
6.6 Products of almost contact manifolds 108
6.7 Examples 111
6.8 Some early topology 117
Curvature of Contact Metric Manifolds 121
7.1 Basic curvature properties 121
7.2 Curvature of contact metric manifolds 126
7.3 The (., µ)-manifolds 133
7.4 Sasakian Einstein manifolds 140
7.5 Locally symmetric contact metric manifolds 142
7.6 Conformally flat contact metric manifolds 143
7.7 f-sectional curvature 147
7.8 Examples of Sasakian space forms 151
7.9 Locally f-symmetric spaces 153
Submanifolds of Kähler and Sasakian Manifolds 160
8.1 Invariant submanifolds 160
8.2 Lagrangian and integral submanifolds 164
Tangent Bundles and Tangent Sphere Bundles 177
9.1 Tangent bundles 177
9.2 Tangent sphere bundles 183
9.3 Geometry of vector bundles 191
9.4 Normal bundles 194
9.5 The geodesic flow on the projectivized tangent bundle 199
Curvature Functionals on Spaces of Associated Metrics 202
10.1 Introduction to critical metric problems 202
10.2 The *-scalar curvature 208
10.3 The integral of Ric(.) 213
10.4 The Webster scalar curvature 219
10.5 A gauge invariant 222
10.6 The Abbena metric as a critical point 224
Negative .-sectional Curvature 226
11.1 Special directions in the contact subbundle 226
11.2 Anosov flows 228
11.3 Conformally Anosov flows 234
Complex Contact Manifolds 239
12.1 Complex contact manifolds and associated metrics 239
12.2 Examples of complex contact manifolds 244
12.3 Normality of complex contact manifolds 256
12.4 GH-sectional curvature 258
12.5 The set of associated metrics and integral functionals 261
12.6 Holomorphic Legendre curves 263
12.7 The Calabi (Veronese) embeddings as integral submanifolds of CP2n+1 266
Additional Topics in Complex Geometry 271
13.1 Partial and holomorphic hyperbolicity 271
13.2 Projectivized holomorphic bundles 274
13.3 The complex geodesic flow 277
13.4 Complex almost contact metric structure on 284
13.5 Special directions on complex contact manifolds and the Lie group SL(2,C) 289
3-Sasakian Manifolds 297
14.1 3-Sasakian manifolds 297
14.2 Integral submanifolds 305
Bibliography 308
Subject Index 339
Author Index 343