Riemannian Geometry of Contact and Symplectic Manifolds (Beitrage Zur Osterreichischen Statistik)

This monograph deals with the Riemannian geometry of both symplectic and contact manifolds, with particular emphasis on the latter. The text is carefully presented. Topics unfold systematically from Chapter 1, which examines the general theory of symplectic manifolds. Principal circle bundles (Chapter 2) are then discussed as a prelude to the Boothby--Wang fibration of a compact regular contact manifold in Chapter 3, which deals with the general theory of contact manifolds. Chapter 4 focuses on the general setting of Riemannian metrics associated with both symplectic and contact structures, and Chapter 5 is devoted to integral submanifolds of the contact subbundle. Topics treated in the subsequent chapters include Sasakian manifolds, the important study of the curvature of contact metric manifolds, submanifold theory in both the K¿hler and Sasakian settings, tangent sphere bundles, curvature functionals, complex contact manifolds and 3 Sasakian manifolds. The book serves both as a general reference for mathematicians to the basic properties of symplectic and contact manifolds and as an excellent resource for graduate students and researchers in the Riemannian geometric arena. The prerequisite for this text is a basic course in Riemannian geometry. Birkhäuser Boston Cover 1 Title Page 4 Copyright 5 Dedication 6 Contents 8 Preface 12 1 Symplectic Manifolds 14 1.1 Definitions and examples 14 1.2 Lagrangian submanifolds 18 1.3 The Darboux-Weinstein theorems 20 1.4 Symplectomorphisms 22 2 Principal S^1-bundles 24 2.1 The set of principal S^1-bundles as a group 24 2.2 Connections on a principal bundle 27 3 Contact Manifolds 30 3.1 Definitions 30 3.2 Examples 33 3.2.1 R^{2n+1} 33 3.2.2 R^{2n+1} x PR^n 34 3.2.3 M^{2n+1} C R^{2n+2} with T_m M^{2n+1} fl {0} = 0 34 3.2.4 Tl M, T1M 35 3.2.5 T*M x1[8 35 3.2.6 T^3 36 3.2.7 T^5 36 3.2.8 Overtwisted contact structures 37 3.2.9 Contact circles 38 3.3 The Boothby-Wang fibration 39 3.4 The Weinstein conjecture 41 4 Associated Metrics 44 4.1 Almost complex and almost contact structures 44 4.2 Polarization and associated metrics 47 4.3 Polarization of metrics as a projection 51 4.3.1 Some linear algebra 52 4.3.2 Results on the set A 54 4.4 Action of symplectic and contact transformations 58 4.5 Examples of almost contact metric manifolds 61 4.5.1 R^{2n+1} 61 4.5.2 M^{2n+1} C M^{2n+2} almost complex 62 4.5.3 S^5 C S^6 63 4.5.4 The Boothby-Wang fibration 65 4.5.5 M^2n X R 66 4.5.6 Parallelizable manifolds 66 5 Integral Submanifolds and Contact Transformations 68 5.1 Integral submanifolds 68 5.2 Contact transformations 70 5.3 Examples of integral submanifolds 72 5.3.1 Sn C Stn+r 72 5.3.2 T2 C S5 72 5.3.3 Legendre curves and Whitney spheres 73 5.3.4 Lift of a Lagrangian submanifold 75 6 Sasakian and Cosymplectic Manifolds 76 6.1 Normal almost contact structures 76 6.2 The tensor field h 80 6.3 Definition of a Sasakian manifold 82 6.4 CR-manifolds 85 6.5 Cosymplectic manifolds and remarks on the Sasakian definition 90 6.6 Products of almost contact manifolds 92 6.7 Examples 94 6.7.1 R^{2n+1} 94 6.7.2 Principal circle bundles 94 6.7.3 A non-normal almost contact structure on S5 96 6.7.4 M^{2n+l} C M^{2n+2} 98 6.7.5 Brieskorn manifolds 98 6.8 Topology 100 7 Curvature of Contact Metric Manifolds 104 7.1 Basic curvature properties 104 7.2 Curvature of contact metric manifolds 108 7.3 0-sectional curvature 123 7.4 Examples of Sasakian space forms 127 7.4.1 S^{2n+1} 127 7.4.2 R^{2n+1} 127 7.4.3 B^n xR 128 7.5 Locally 0-symmetric spaces 128 8 Submanifolds of Kahler and Sasakian Manifolds 134 8.1 Invariant submanifolds 134 8.2 Lagrangian and integral submanifolds 137 8.3 Legendre curves 146 9 Tangent Bundles and Tangent Sphere Bundles 150 9.1 Tangent bundles 150 9.2 Tangent sphere bundles 155 9.3 Geometry of vector bundles 161 9.4 Normal bundles 163 10 Curvature Functionals on Spaces of Associated Metrics 170 10.1 Introduction to critical metric problems 170 10.2 The *-scalar curvature 175 10.3 The integral of Ric(h) 179 10.4 The Webster scalar curvature 183 10.5 A gauge invariant 186 10.6 The Abbena metric as a critical point 187 11 Negative c-sectional Curvature 190 11.1 Special Directions in the contact subbundle 190 11.2 Anosov flows 191 11.3 Conformally Anosov flows 197 12 Complex Contact Manifolds 202 12.1 Complex contact manifolds and associated metrics 202 12.2 Examples of complex contact manifolds 206 12.2.1 Complex Heisenberg group 206 12.2.2 Odd-dimensional complex projective space 207 12.2.3 Twistor spaces 209 12.2.4 The complex Boothby-Wang fibration 211 12.2.5 3-dimensional homogeneous examples 213 12.2.6 C^{n+1} X CP^n(16) 213 12.3 Normality of complex contact manifolds 215 12.4 GH-sectional curvature 217 12.5 The set of associated metrics and integral functionals 219 12.6 Holomorphic Legendre curves 222 12.7 The Calabi (Veronese) imbeddings as integral submanifolds of CP^{2n+1} 225 13 3-Sasakian Manifolds 228 13.1 3-Sasakian manifolds 228 13.2 Integral submanifolds 236 Bibliography 240 Subject Index 266 Author Index 270 Back Cover 274 0817642617,9780817642617,3764342617,9783764342616 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 The author's lectures,'Contact Manifolds in Riemannian Geometry,'volume 509 (1976), in the Springer-Verlag Lecture Notes in Mathematics series have been out of print for some time and it seems appropriate that an expanded version of this material should become available. The present text deals with the Riemannian geometry of both symplectic and contact manifolds, although the book is more contact than symplectic. This work is based on the recent research of the author, his students, colleagues, and other scholars, the author's graduate courses at Michigan State University and the earlier lecture notes. Chapter 1 presents the general theory of symplectic manifolds. Principal circle bundles are then discussed in Chapter 2 as a prelude to the Boothby­ Wang fibration of a compact regular contact manifold in Chapter 3, which deals with the general theory of contact manifolds. Chapter 4 focuses on Rie­ mannian metrics associated to symplectic and contact structures. Chapter 5 is devoted to integral submanifolds of the contact subbundle. In Chapter 6 we discuss the normality of almost contact structures, Sasakian manifolds, K­ contact manifolds, the relation of contact metric structures and CR-structures, and cosymplectic structures. Chapter 7 deals with the important study of the curvature of a contact metric manifold. In Chapter 8 we give a selection of results on submanifolds of Kahler and Sasakian manifolds, including an illus­ tration of the technique of A. Ros in a theorem of F. Urbano on compact minimal Lagrangian sub manifolds in cpn. To set the stage for our development, we begin this book with a treatment of the basic features of symplectic geometry.