Riemannian Geometry of Contact and Symplectic Manifolds

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. 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 Front Matter....Pages i-xii Symplectic Manifolds....Pages 1-10 Principal S 1 -bundles....Pages 11-16 Contact Manifolds....Pages 17-29 Associated Metrics....Pages 31-53 Integral Submanifolds and Contact Transformations....Pages 55-62 Sasakian and Cosymplectic Manifolds....Pages 63-89 Curvature of Contact Metric Manifolds....Pages 91-120 Submanifolds of Kähler and Sasakian Manifolds....Pages 121-135 Tangent Bundles and Tangent Sphere Bundles....Pages 137-155 Curvature Functionals on Spaces of Associated Metrics....Pages 157-175 Negative ξ -sectional Curvature....Pages 177-187 Complex Contact Manifolds....Pages 189-214 3-Sasakian Manifolds....Pages 215-225 Back Matter....Pages 227-260