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Riemannian Geometry of Contact and Symplectic Manifolds

  • Book
  • © 2002

Overview

Authors:
  1. David E. Blair

    1. Department of Mathematics, Michigan State University, East Lansing, USA

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Part of the book series: Progress in Mathematics (PM, volume 203)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xii
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  2. Symplectic Manifolds

    • David E. Blair
    Pages 1-10

  3. Principal S 1-bundles

    • David E. Blair
    Pages 11-16

  4. Contact Manifolds

    • David E. Blair
    Pages 17-29

  5. Associated Metrics

    • David E. Blair
    Pages 31-53

  6. Integral Submanifolds and Contact Transformations

    • David E. Blair
    Pages 55-62

  7. Sasakian and Cosymplectic Manifolds

    • David E. Blair
    Pages 63-89

  8. Curvature of Contact Metric Manifolds

    • David E. Blair
    Pages 91-120

  9. Submanifolds of Kähler and Sasakian Manifolds

    • David E. Blair
    Pages 121-135

  10. Tangent Bundles and Tangent Sphere Bundles

    • David E. Blair
    Pages 137-155

  11. Curvature Functionals on Spaces of Associated Metrics

    • David E. Blair
    Pages 157-175

  12. Negative Îľ-sectional Curvature

    • David E. Blair
    Pages 177-187

  13. Complex Contact Manifolds

    • David E. Blair
    Pages 189-214

  14. 3-Sasakian Manifolds

    • David E. Blair
    Pages 215-225

  15. Back Matter

    Pages 227-260
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Keywords

About this book

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.

Reviews

"The book . . . supplies a lot of examples, and includes many recent results. It can be used either as an introduction to the subject or as a reference for students and researchers . . . [it] gives a clear and complete account of the main ideas . . . and studies a vast amount of related subjects such as integral submanifolds, symplectic structure of tangent bundles, curvature of contact metric manifolds and curvature functionals on spaces of associated metrics."   —Mathematical Reviews

"Several examples accompany almost all chapters, in the first book in which the geometry of complex contact manifold[s] is presented. Always, a correct balance between theory and examples is maintained. Also the author gives detail and instructive proofs for basic results and states, without proofs, a great number of interesting results in addition to the corresponding references...[T]his is a pleasant and useful book and all geometers will profit [from] reading it. They can use it for advanced courses, for thesis topics as well as for references. Beginners will find in it an attractive [table of] contents and useful ideas for pursuing their studies."   —Memoriile Sectiilor Stiintifice

Authors and Affiliations

  • Department of Mathematics, Michigan State University, East Lansing, USA

    David E. Blair

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