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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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TL;DR: In this paper, the authors extend the notion of generalized Sasakian space form to the semi-Riemannian setting, and consider several interesting cases and give examples of them all.
Abstract: In this paper, we extend the notion of generalized Sasakian space form to the semi-Riemannian setting. We consider several interesting cases and we give examples of them all. We also study their structures.
14 citations
TL;DR: In this article, the Levi-Kahler quotient of toric CR manifolds has been studied in arbitrary codimension, and a process called the Levi Kullback quotient is introduced for constructing Kahler metrics from CR structures with a transverse torus action.
Abstract: We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.
14 citations
Cites background from "Riemannian Geometry of Contact and ..."
...(see [24] and [44]) that the Chern–Moser tensor of (Ni,Ji,Di) may be computed from the horizontal part of the curvature of the LEVI–KAHLER REDUCTION AND TORIC GEOMETRY 31¨ Tanaka connection (see e.g. [14]) associated to any contact form αcompatible with the (codimension 1) CR structure (D,J). The Chern–Moser tensor does not depend upon the chosen compatible contact structure (it is a CR invariant). If...
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TL;DR: In this article, a new second variation formula was derived for minimal Legendrian submanifolds in Sasakian manifolds, and the notion of the Legendrian stability was introduced.
Abstract: First, we derive a new second variation formula which holds for minimal Legendrian submanifolds in Sasakian manifolds. Using this, we prove that any minimal Legendrian submanifold in an η -Einstein Sasakian manifold with “nonpositive” η -Ricci constant is stable. Next we introduce the notion of the Legendrian stability of minimal Legendrian submanifolds in Sasakian manifolds. Using our second variation formula, we find a general criterion for the Legendrian stability of minimal Legendrian submanifolds in η -Einstein Sasakian manifolds with “positive” η -Ricci constant.
14 citations
TL;DR: In this paper, a 3-dimensional paraSasakian manifold and a conformally flat K-paracontact manifold were studied and it was shown that the conditions Einstein, conformal flat, semi-symmetric, and Ricci semi symmetric are all equivalent.
Abstract: The purpose of this paper is to study 3-dimensional paraSasakian manifold and conformally flat K-paracontact manifold. Moreover, we show that in a K-paracontact manifold the conditions Einstein, conformally flat, semi-symmetric and Ricci semi-symmetric are all equivalent. Finally, compact regular 3-dimensional paraSasakian and conformally flat K-paracontact manifolds are studied.
14 citations
TL;DR: In this paper, almost paracontact Riemannian manifolds of the lowest dimension 3 were constructed on a family of Lie groups and the obtained manifolds were studied. And the Curvature properties of these manifolds are investigated.
Abstract: Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of the obtained results.
14 citations