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Riemannian Geometry of Contact and Symplectic Manifolds

08 Jan 2002-
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-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 Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index

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Journal ArticleDOI
TL;DR: In this paper, a complete solution to the contact equivalence problem for a class of toric contact structures, Y p;q, discovered by physicists by showing that Y p 0 ;q 0 are inequivalent as contact structures if and only if p6 p 0.
Abstract: I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold S 2 S 3 . In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, Y p;q , discovered by physicists by showing that Y p;q and Y p 0 ;q 0 are inequivalent as contact structures if and only if p6 p 0 .

63 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...In this section we give a very brief review of contact geometry referring to the books [50, 12, 22, 11] for details....

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Journal ArticleDOI
TL;DR: In this article, the complete classification of homogeneous paracontact metric manifolds is obtained, in the symmetric case, such a manifold is either flat or of constant sectional curvature −1, and in the non-symmetric case it is a Lie group equipped with a left-invariant metric structure.
Abstract: The complete classification of three-dimensional homogeneous paracontact metric manifolds is obtained. In the symmetric case, such a manifold is either flat or of constant sectional curvature −1. In the non-symmetric case, it is a Lie group equipped with a left-invariant paracontact metric structure.

63 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied curves and surfaces in 3-dimensional contact manifolds whose mean curvature vector is in the kernel of certain elliptic dieren tial operators.
Abstract: Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated. Introduction. This paper concerns curves and surfaces in 3-dimen- sional contact manifolds whose mean curvature vector eld is in the kernel of certain elliptic dieren tial operators. First we study submanifolds whose mean curvature vector eld is in the kernel of the Laplacian (submanifolds with harmonic mean curvature vector elds). The study of such submanifolds is inspired by a conjecture of Bang-yen Chen (14):

62 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...1 Contact manifolds We begin by recalling fundamental ingredients of contact Riemannian geometry from [7]....

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Journal ArticleDOI
TL;DR: In this article, the authors discuté l'integrabilite des varietes de Jacobi par des groupoides de contact and showed that le point de vue des structures of Jacobi apporte a la geometrie de Poisson.
Abstract: Nous discutons l'integrabilite des varietes de Jacobi par des groupoides de contact. Nous considerons ensuite ce que le point de vue des structures de Jacobi apporte a la geometrie de Poisson. En particulier, en utilisant les groupoides de contacts, nous prouvons un theoreme a la Kostant sur la prequantization des groupoides symplectiques. Ce theoreme repond a une question posee par Weinstein et Xu. Nous utilisons les methodes de Crainic-Fernandes sur les A-paths et les group(oid)es de monodromie d'algebroides. En particulier, la plupart des resultats que nous obtenons sont valides dans le cas non-integrable.

61 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...[1]): θ and θ′ are equivalent if θ′ = τθ for some non-vanishing function τ ....

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  • ...Kostant’s theorem (sometimes also attributed to Kobayashi [1] or to Souriau) says that this is possible if and only if ω is integral....

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Journal ArticleDOI
TL;DR: In this article, the hermitian analog of Aleksandrov's area measures of convex bodies is investigated and a characterization of those area measures which arise as the first variation of unitarily invariant valuations is established.
Abstract: The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily invariant area measures is described explicitly.

60 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...The Rumin differential operator [50] is defined on a general contact manifold, but for our purposes it is sufficient to consider it only in the special case of the sphere bundle (we refer to [23] for all notions from contact geometry)....

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