This book, "Riemannian Geometry of Contact and Symplectic Manifolds" by David E. Blair, is a comprehensive treatment of the Riemannian geometry of contact and symplectic manifolds. The content is structured into several chapters, each focusing on different aspects of these geometric structures: 1. **Symplectic Manifolds**: Introduces the basic definitions, examples, and theorems related to symplectic manifolds, including Lagrangian submanifolds and symplectomorphisms. 2. **Principal \(S^1\)-bundles**: Discusses the structure of principal \(S^1\)-bundles and connections on these bundles. 3. **Contact Manifolds**: Explores the definitions, examples, and the Boothby–Wang fibration of compact regular contact manifolds. 4. **Associated Metrics**: Focuses on almost complex and almost contact structures, polarization, and associated metrics. 5. **Integral Submanifolds and Contact Transformations**: Covers integral submanifolds and the action of contact transformations. 6. **Sasakian and Cosymplectic Manifolds**: Discusses normal almost contact structures, Sasakian manifolds, CR-manifolds, and cosymplectic manifolds. 7. **Curvature of Contact Metric Manifolds**: Studies basic curvature properties and the curvature of contact metric manifolds. 8. **Submanifolds of Kähler and Sasakian Manifolds**: Provides results on invariant submanifolds, Lagrangian and integral submanifolds, and Legendre curves. 9. **Tangent Bundles and Tangent Sphere Bundles**: Examines the geometry of vector bundles and normal bundles. 10. **Curvature Functionals on Spaces of Associated Metrics**: Introduces curvature functionals and their critical point conditions. 11. **Negative \(\xi\)-sectional Curvature**: Discusses special directions in the contact subbundle and their relations to Anosov and conformally Anosov flows. 12. **Complex Contact Manifolds**: Explores complex contact manifolds and their associated metrics, including examples and normality conditions. 13. **3-Sasakian Manifolds**: Provides a brief treatment of 3-Sasakian manifolds. The book aims to provide a detailed introduction to the Riemannian geometry of contact and symplectic manifolds, while also serving as a useful reference for recent research in the area. It includes extensive bibliographies and acknowledges the contributions of several scholars who reviewed parts of the manuscript.This book, "Riemannian Geometry of Contact and Symplectic Manifolds" by David E. Blair, is a comprehensive treatment of the Riemannian geometry of contact and symplectic manifolds. The content is structured into several chapters, each focusing on different aspects of these geometric structures: 1. **Symplectic Manifolds**: Introduces the basic definitions, examples, and theorems related to symplectic manifolds, including Lagrangian submanifolds and symplectomorphisms. 2. **Principal \(S^1\)-bundles**: Discusses the structure of principal \(S^1\)-bundles and connections on these bundles. 3. **Contact Manifolds**: Explores the definitions, examples, and the Boothby–Wang fibration of compact regular contact manifolds. 4. **Associated Metrics**: Focuses on almost complex and almost contact structures, polarization, and associated metrics. 5. **Integral Submanifolds and Contact Transformations**: Covers integral submanifolds and the action of contact transformations. 6. **Sasakian and Cosymplectic Manifolds**: Discusses normal almost contact structures, Sasakian manifolds, CR-manifolds, and cosymplectic manifolds. 7. **Curvature of Contact Metric Manifolds**: Studies basic curvature properties and the curvature of contact metric manifolds. 8. **Submanifolds of Kähler and Sasakian Manifolds**: Provides results on invariant submanifolds, Lagrangian and integral submanifolds, and Legendre curves. 9. **Tangent Bundles and Tangent Sphere Bundles**: Examines the geometry of vector bundles and normal bundles. 10. **Curvature Functionals on Spaces of Associated Metrics**: Introduces curvature functionals and their critical point conditions. 11. **Negative \(\xi\)-sectional Curvature**: Discusses special directions in the contact subbundle and their relations to Anosov and conformally Anosov flows. 12. **Complex Contact Manifolds**: Explores complex contact manifolds and their associated metrics, including examples and normality conditions. 13. **3-Sasakian Manifolds**: Provides a brief treatment of 3-Sasakian manifolds. The book aims to provide a detailed introduction to the Riemannian geometry of contact and symplectic manifolds, while also serving as a useful reference for recent research in the area. It includes extensive bibliographies and acknowledges the contributions of several scholars who reviewed parts of the manuscript.