Book•
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 article, it was shown that there is no totally contact geodesic proper slant light-like submanifold in indefinite Kenmotsu manifolds other than a geodesically-convex geodesics proper slantsic submanivold.
Abstract: In this paper, we study totally contact umbilical slant lightlike submanifolds of indefinite Kenmotsu manifolds. We prove that there does not exist totally contact umbilical proper slant lightlike submanifold in indefinite Kenmotsu manifolds other than totally contact geodesic proper slant lightlike submanifold. We also prove that there does not exist totally contact umbilical proper slant lightlike submanifold of indefinite Kenmotsu space forms. Finally, we give some characterization theorems on minimal slant lightlike submanifolds of indefinite Kenmotsu manifolds.
3 citations
TL;DR: In this article, a special type of semi-symmetric non-metric φ-connection on a Kenmotsu manifold is studied, and it is shown that if the curvature tensor of the manifold admits a special semi-sysmmetric nonsmidth-connection, then the manifold is locally isometric to the hyperbolic space H(−1).
Abstract: The object of the present paper is to study a special type of semi-symmetric non-metric φ-connection on a Kenmotsu manifold. It is shown that if the curvature tensor of Kenmotsu manifolds admitting a special type of semi-symmetric non-metric φ-connection ∇̄ vanishes, then the Kenmotsu manifold is locally isometric to the hyperbolic space H(−1). Beside these, we consider Weyl conformal curvature tensor of a Kenmotsu manifold with respect to the semi-symmetric non-metric φ-connection. Among other results, we prove that the Weyl conformal curvature tensor with respect to the Levi-Civita connection and the semisymmetric non-metric φ-connection are equivalent. Moreover, we deal with φ-Weyl semi-symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric φ-connection. Finally, an illustrative example is given to verify our result. AMS Mathematics Subject Classification (2010): 53C15; 53C25
3 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Let M be an (2n+ 1)-dimensional almost contact metric manifold with an almost contact metric structure (φ, ξ, η, g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and the Riemannian metric g on M satisfying ([4], [5])...
[...]
3 citations
TL;DR: In this paper, the divergence of any vector field is invariant under a D-homothetically deformed K-contact Ricci almost soliton with the same associated vector field.
Abstract: If a K-contact manifold (M, g) and a D-homothetically deformed K-contact manifold
$$(M,{\bar{g}})$$
are both Ricci almost solitons with the same associated vector field V, then we show (i) that (M, g) and (
$$M, {\bar{g}}$$
) are both D-homothetically fixed
$$\eta $$
-Einstein Ricci solitons, and (ii) V preserves
$$\phi $$
. We also show that, if the associated vector field V of a complete K-contact Ricci almost soliton (M, g, V) is a projective vector field, then V is Killing and (M, g) is compact Sasakian and shrinking. Finally, we show that the divergence of any vector field is invariant under a D-homothetic deformation.
3 citations
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TL;DR: In this paper, the Buhovsky-Seyfaddini-Viterbo uniqueness theorem for topological strictly contact Hamiltonians was shown to be equivalent to the one-to-one correspondence between topological flows and Hamiltonians.
Abstract: Let $M$ be a smooth, closed and connected manifold carrying a cooriented contact structure. Based on the approach of S. M\"uller and Y.-G. Oh we construct a metric topology on the space of strictly contact isotopies of $M$ and use it to define a certain completion that we call the collection of topological strictly contact isotopies. This collection of isotopies is a topological group. The set of time-one maps of these isotopies also forms a topological group satisfying a transformation law that restricts to the usual transformation law for strictly contact diffeomorphisms.
Suppose the contact form on $M$ is regular. We prove that the topological strictly contact isotopy associated to a topological contact Hamiltonian function is unique and consequently we extend the group laws for smooth basic contact Hamiltonian functions to the collection of topological contact Hamiltonian functions. We next prove the Buhovsky-Seyfaddini-Viterbo uniqueness theorem for contact Hamiltonians, completing the one-to-one correspondence between topological flows and Hamiltonians. Furthermore the group of strictly contact homeomorphisms is a central extension of the group of Hamiltonian homeomorphisms of the quotient of $M$ by the flow of the Reeb field of the regular contact form. Finally, using this $S^1$-extension, we prove the analogue of a theorem of M\"uller for Hamiltonian diffeomorphisms that says the group of strictly contact homeomorphisms is independent of a certain choice of norm used in the construction.
2 citations