Blair Riemannian Geometry of Contact and Symplectic Manifolds

2. Auflage 2010
ISBN: 978-0-8176-4959-3
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 343 Seiten

Reihe: Progress in Mathematics

This second edition, divided into fourteen chapters, presents a comprehensive treatment of contact and symplectic manifolds from the Riemannian point of view. The monograph examines the basic ideas in detail and provides many illustrative examples for the reader.Riemannian Geometry of Contact and Symplectic Manifolds, Second Edition provides new material in most chapters, but a particular emphasis remains on contact manifolds. Researchers, mathematicians, and graduate students in contact and symplectic manifold theory and in Riemannian geometry will benefit from this work. A basic course in Riemannian geometry is a prerequisite.

Autoren/Hrsg.

Blair, David E.

Inhaltsverzeichnis

1;Preface to Second Edition;7
2;Preface to the First Edition;8
3;Contents;10
4;Symplectic Manifolds;15
4.1;1.1 Definitions and examples;15
4.2;1.2 Lagrangian submanifolds;20
4.3;1.3 The Darboux–Weinstein theorems;23
4.4;1.4 Symplectomorphisms;25
5;Principal S1-bundles;28
5.1;2.1 The set of principal S1-bundles as a group;28
5.2;2.2 Connections on a principal bundle;32
6;Contact Manifolds;35
6.1;3.1 Definitions;35
6.2;3.2 Examples;39
6.3;3.3 The Boothby–Wang fibration;48
6.4;3.4 The Weinstein conjecture;50
7;Associated Metrics;53
7.1;4.1 Almost complex and almost contact structures;53
7.2;4.2 Polarization and associated metrics;57
7.3;4.3 Polarization of metrics as a projection;61
7.4;4.4 Action of symplectic and contact transformations;69
7.5;4.5 Examples of almost contact metric manifolds;72
8;Integral Submanifolds and Contact Transformations;80
8.1;5.1 Integral submanifolds;80
8.2;5.2 Contact transformations;82
8.3;5.3 Examples of integral submanifolds;85
9;Sasakian and Cosymplectic Manifolds;90
9.1;6.1 Normal almost contact structures;90
9.2;6.2 The tensor field;94
9.3;6.3 Definition of a Sasakian manifold;97
9.4;6.4 CR-manifolds;100
9.5;6.5 Cosymplectic manifolds and remarks on the Sasakian definition;106
9.6;6.6 Products of almost contact manifolds;108
9.7;6.7 Examples;111
9.8;6.8 Some early topology;117
10;Curvature of Contact Metric Manifolds;121
10.1;7.1 Basic curvature properties;121
10.2;7.2 Curvature of contact metric manifolds;126
10.3;7.3 The (., µ)-manifolds;133
10.4;7.4 Sasakian Einstein manifolds;140
10.5;7.5 Locally symmetric contact metric manifolds;142
10.6;7.6 Conformally flat contact metric manifolds;143
10.7;7.7 f-sectional curvature;147
10.8;7.8 Examples of Sasakian space forms;151
10.9;7.9 Locally f-symmetric spaces;153
11;Submanifolds of Kähler and Sasakian Manifolds;160
11.1;8.1 Invariant submanifolds;160
11.2;8.2 Lagrangian and integral submanifolds;164
12;Tangent Bundles and Tangent Sphere Bundles;177
12.1;9.1 Tangent bundles;177
12.2;9.2 Tangent sphere bundles;183
12.3;9.3 Geometry of vector bundles;191
12.4;9.4 Normal bundles;194
12.5;9.5 The geodesic flow on the projectivized tangent bundle;199
13;Curvature Functionals on Spaces of Associated Metrics;202
13.1;10.1 Introduction to critical metric problems;202
13.2;10.2 The *-scalar curvature;208
13.3;10.3 The integral of Ric(.);213
13.4;10.4 The Webster scalar curvature;219
13.5;10.5 A gauge invariant;222
13.6;10.6 The Abbena metric as a critical point;224
14;Negative .-sectional Curvature;226
14.1;11.1 Special directions in the contact subbundle;226
14.2;11.2 Anosov flows;228
14.3;11.3 Conformally Anosov flows;234
15;Complex Contact Manifolds;239
15.1;12.1 Complex contact manifolds and associated metrics;239
15.2;12.2 Examples of complex contact manifolds;244
15.3;12.3 Normality of complex contact manifolds;256
15.4;12.4 GH-sectional curvature;258
15.5;12.5 The set of associated metrics and integral functionals;261
15.6;12.6 Holomorphic Legendre curves;263
15.7;12.7 The Calabi (Veronese) embeddings as integral submanifolds of CP2n+1;266
16;Additional Topics in Complex Geometry;271
16.1;13.1 Partial and holomorphic hyperbolicity;271
16.2;13.2 Projectivized holomorphic bundles;274
16.3;13.3 The complex geodesic flow;277
16.4;13.4 Complex almost contact metric structure on;284
16.5;13.5 Special directions on complex contact manifolds and the Lie group SL(2,C);289
17;3-Sasakian Manifolds;297
17.1;14.1 3-Sasakian manifolds;297
17.2;14.2 Integral submanifolds;305
18;Bibliography;308
19;Subject Index;339
20;Author Index;343