Riemannian Geometry of Contact and Symplectic Manifolds

Riemannian Geometry of Contact and Symplectic Manifolds

2002 | David E. Blair
This book, "Riemannian Geometry of Contact and Symplectic Manifolds" by David E. Blair, is a comprehensive treatment of the Riemannian geometry of both symplectic and contact manifolds. While the book is more focused on contact manifolds, it also covers symplectic geometry. The content is based on the author's research, his students' work, and his graduate courses at Michigan State University. The book is structured into 13 chapters, each covering various aspects of symplectic and contact geometry. Chapter 1 introduces symplectic manifolds, while Chapter 2 discusses principal circle bundles, which are used in the Boothby-Wang fibration of compact regular contact manifolds. Chapter 3 presents the general theory of contact manifolds. Chapter 4 focuses on Riemannian metrics associated with symplectic and contact structures. Chapter 5 deals with integral submanifolds of the contact subbundle. Chapter 6 discusses the normality of almost contact structures, Sasakian manifolds, K-contact manifolds, and cosymplectic structures. Chapter 7 explores the curvature of contact metric manifolds. Chapter 8 presents results on submanifolds of Kähler and Sasakian manifolds. Chapter 9 discusses the symplectic structure of tangent bundles and the contact structure of tangent sphere bundles. Chapter 10 studies curvature functionals on spaces of associated metrics. Chapter 11 examines negative sectional curvature and its relation to Anosov flows. Chapter 12 covers complex contact manifolds. Chapter 13 provides a brief treatment of 3-Sasakian manifolds. The book provides detailed proofs of basic properties and states many results, while avoiding an encyclopedic approach. It includes an extensive bibliography and is intended as both an introduction to the subject and a reference for recent research. The author thanks several individuals for their contributions to the book's production.This book, "Riemannian Geometry of Contact and Symplectic Manifolds" by David E. Blair, is a comprehensive treatment of the Riemannian geometry of both symplectic and contact manifolds. While the book is more focused on contact manifolds, it also covers symplectic geometry. The content is based on the author's research, his students' work, and his graduate courses at Michigan State University. The book is structured into 13 chapters, each covering various aspects of symplectic and contact geometry. Chapter 1 introduces symplectic manifolds, while Chapter 2 discusses principal circle bundles, which are used in the Boothby-Wang fibration of compact regular contact manifolds. Chapter 3 presents the general theory of contact manifolds. Chapter 4 focuses on Riemannian metrics associated with symplectic and contact structures. Chapter 5 deals with integral submanifolds of the contact subbundle. Chapter 6 discusses the normality of almost contact structures, Sasakian manifolds, K-contact manifolds, and cosymplectic structures. Chapter 7 explores the curvature of contact metric manifolds. Chapter 8 presents results on submanifolds of Kähler and Sasakian manifolds. Chapter 9 discusses the symplectic structure of tangent bundles and the contact structure of tangent sphere bundles. Chapter 10 studies curvature functionals on spaces of associated metrics. Chapter 11 examines negative sectional curvature and its relation to Anosov flows. Chapter 12 covers complex contact manifolds. Chapter 13 provides a brief treatment of 3-Sasakian manifolds. The book provides detailed proofs of basic properties and states many results, while avoiding an encyclopedic approach. It includes an extensive bibliography and is intended as both an introduction to the subject and a reference for recent research. The author thanks several individuals for their contributions to the book's production.
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