Riemannian Geometry of Contact and Symplectic Manifolds

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 12
1.1 Definitions and examples......Page 14
1.2 Lagrangian submanifolds......Page 18
1.3 The Darboux-Weinstein theorems......Page 20
1.4 Symplectomorphisms......Page 22
2.1 The set of principal S^1-bundles as a group......Page 24
2.2 Connections on a principal bundle......Page 27
3.1 Definitions......Page 30
3.2.1 R^{2n+1}......Page 33
3.2.3 M^{2n+1} C R^{2n+2} with T_m M^{2n+1} fl {0} = 0......Page 34
3.2.5 T*M x1[8......Page 35
3.2.7 T^5......Page 36
3.2.8 Overtwisted contact structures......Page 37
3.2.9 Contact circles......Page 38
3.3 The Boothby-Wang fibration......Page 39
3.4 The Weinstein conjecture......Page 41
4.1 Almost complex and almost contact structures......Page 44
4.2 Polarization and associated metrics......Page 47
4.3 Polarization of metrics as a projection......Page 51
4.3.1 Some linear algebra......Page 52
4.3.2 Results on the set A......Page 54
4.4 Action of symplectic and contact transformations......Page 58
4.5.1 R^{2n+1}......Page 61
4.5.2 M^{2n+1} C M^{2n+2} almost complex......Page 62
4.5.3 S^5 C S^6......Page 63
4.5.4 The Boothby-Wang fibration......Page 65
4.5.6 Parallelizable manifolds......Page 66
5.1 Integral submanifolds......Page 68
5.2 Contact transformations......Page 70
5.3.2 T2 C S5......Page 72
5.3.3 Legendre curves and Whitney spheres......Page 73
5.3.4 Lift of a Lagrangian submanifold......Page 75
6.1 Normal almost contact structures......Page 76
6.2 The tensor field h......Page 80
6.3 Definition of a Sasakian manifold......Page 82
6.4 CR-manifolds......Page 85
6.5 Cosymplectic manifolds and remarks on the Sasakian definition......Page 90
6.6 Products of almost contact manifolds......Page 92
6.7.2 Principal circle bundles......Page 94
6.7.3 A non-normal almost contact structure on S5......Page 96
6.7.5 Brieskorn manifolds......Page 98
6.8 Topology......Page 100
7.1 Basic curvature properties......Page 104
7.2 Curvature of contact metric manifolds......Page 108
7.3 0-sectional curvature......Page 123
7.4.2 R^{2n+1}......Page 127
7.5 Locally 0-symmetric spaces......Page 128
8.1 Invariant submanifolds......Page 134
8.2 Lagrangian and integral submanifolds......Page 137
8.3 Legendre curves......Page 146
9.1 Tangent bundles......Page 150
9.2 Tangent sphere bundles......Page 155
9.3 Geometry of vector bundles......Page 161
9.4 Normal bundles......Page 163
10.1 Introduction to critical metric problems......Page 170
10.2 The *-scalar curvature......Page 175
10.3 The integral of Ric(h)......Page 179
10.4 The Webster scalar curvature......Page 183
10.5 A gauge invariant......Page 186
10.6 The Abbena metric as a critical point......Page 187
11.1 Special Directions in the contact subbundle......Page 190
11.2 Anosov flows......Page 191
11.3 Conformally Anosov flows......Page 197
12.1 Complex contact manifolds and associated metrics......Page 202
12.2.1 Complex Heisenberg group......Page 206
12.2.2 Odd-dimensional complex projective space......Page 207
12.2.3 Twistor spaces......Page 209
12.2.4 The complex Boothby-Wang fibration......Page 211
12.2.6 C^{n+1} X CP^n(16)......Page 213
12.3 Normality of complex contact manifolds......Page 215
12.4 GH-sectional curvature......Page 217
12.5 The set of associated metrics and integral functionals......Page 219
12.6 Holomorphic Legendre curves......Page 222
12.7 The Calabi (Veronese) imbeddings as integral submanifolds of CP^{2n+1}......Page 225
13.1 3-Sasakian manifolds......Page 228
13.2 Integral submanifolds......Page 236
Bibliography......Page 240
Subject Index......Page 266
Author Index......Page 270
Back Cover......Page 274