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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 studied the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form and derived the equivalence between a holomorphic ODE system and an associated real ODE systems.
Abstract: The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic last multipliers with real last multipliers. Also, we consider some applications of our study for holomorphic gradient vector fields on holomorphic Riemannain manifolds as well as for holomorphic Hamiltonian vector fields and holomorphic Poisson bivector fields on holomorphic Poisson manifolds.
3 citations
TL;DR: In this article, the authors studied the critical point equation conjecture on almost Kenmotsu manifolds and proved that if a three-dimensional almost-k,\mu)-manifold satisfies the conjecture, then the manifold is either locally isometric to the product space or is a kinematic manifold.
Abstract: We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ or the manifold is Kenmotsu manifold. Further, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the $CPE$ conjecture, then the manifold is Einstein.
3 citations
TL;DR: In this paper, the authors used root decomposition techniques to classify the complex contact Lie groups such that the Reeb vector field action on the Lie algebra is diagonalizable, and these groups turn out to be isomorphic on a Lie algebra level to a particular type of generalized Heisenberg groups, namely the semi-direct product C 2n×ΩC, where Ω is the standard symplectic 2-form on C2n.
Abstract: In this paper, we use root decomposition techniques to classify the complex contact Lie groups such that the Reeb vector field action on the Lie algebra is diagonalizable. These groups turn out to be isomorphic on the Lie algebra level to a particular type of generalized Heisenberg groups, namely the semi-direct product C2n×ΩC, where Ω is the standard symplectic 2-form on C2n.
3 citations
Cites background from "Riemannian Geometry of Contact and ..."
...For a general exposition of complex contact manifolds, see Chapter 12 of [1]....
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TL;DR: In this paper, the authors studied contact metric hypersurfaces of a Bochner-Kaehler manifold and obtained the following two results: (1) a contact metric constant mean curvature (CMC) hypersurface of a BK manifold is a (k, μ)-contact manifold, and (2) if M is a compact contact metric CMC hypersusurface with a conformal vector field V that is neither tangential nor normal anywhere, then it is totally umbilical and Sasakian, and under certain conditions on V, is isometric to
Abstract: We have studied contact metric hypersurfaces of a Bochner–Kaehler manifold and obtained the following two results: (1) a contact metric constant mean curvature (CMC) hypersurface of a Bochner–Kaehler manifold is a (k, μ)-contact manifold, and (2) if M is a compact contact metric CMC hypersurface of a Bochner–Kaehler manifold with a conformal vector field V that is neither tangential nor normal anywhere, then it is totally umbilical and Sasakian, and under certain conditions on V, is isometric to a unit sphere.
3 citations
Additional excerpts
...For details we refer to Blair [1]....
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