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**Riemannian geometry of contact and symplectic manifolds.**
*(English)*
Zbl 1011.53001

Progress in Mathematics (Boston, Mass.). 203. Boston, MA: Birkhäuser. xii, 260 p. (2002).

As the author mentions in his preface, his lectures “Contact manifolds in Riemannian geometry“, in the Springer-Verlag Lecture Notes in Mathematics series Vol. 509 (1976; Zbl 0319.53026) 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 …and the author’s graduate courses at Michigan State University”.

The main subjects of this text which can serve as a general references for basic properties of Riemannian manifolds possessing an additional contact or symplectic structure are demonstrated by the list of contents presented below:

1. Symplectic manifolds 2. Principal \(S^1\)-bundles. 3. Contact manifolds. 4. Associated metrics. 5. Integral submanifolds and contact transformations. 6. Sasakian and cosymplectic manifolds. 7. Curvature of contact metric manifolds. 8. Submanifolds of Kähler and Sasakian manifolds. 9. Tangent bundles and tangent sphere bundles. 10. Curvature functionals on spaces of associated metrics. 11. Negative \(\xi\)-sectional curvature. 12. Complex contact manifolds. 13. \(3\)-Sasakian manifolds.

The main subjects of this text which can serve as a general references for basic properties of Riemannian manifolds possessing an additional contact or symplectic structure are demonstrated by the list of contents presented below:

1. Symplectic manifolds 2. Principal \(S^1\)-bundles. 3. Contact manifolds. 4. Associated metrics. 5. Integral submanifolds and contact transformations. 6. Sasakian and cosymplectic manifolds. 7. Curvature of contact metric manifolds. 8. Submanifolds of Kähler and Sasakian manifolds. 9. Tangent bundles and tangent sphere bundles. 10. Curvature functionals on spaces of associated metrics. 11. Negative \(\xi\)-sectional curvature. 12. Complex contact manifolds. 13. \(3\)-Sasakian manifolds.

Reviewer: Iskander A.Taimanov (Novosibirsk)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53Dxx | Symplectic geometry, contact geometry |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53D35 | Global theory of symplectic and contact manifolds |

53D05 | Symplectic manifolds (general theory) |

53D10 | Contact manifolds (general theory) |