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 paper, the authors prove a local splitting theorem similar to those appearing in Poisson geometry for generalized contact bundles, where a generalized contact bundle is either the product of a contact and a complex manifold or a symplectic manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.
Abstract: Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.
6 citations
01 Jan 2020
TL;DR: In this paper, the generalized Tanaka-Webster connection in a contact Lorentzian manifold was shown to be a necessary and sufficient condition for the ∇ ^ -geodesic.
Abstract: In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .
6 citations
TL;DR: In this paper, the projection of the image of concircular curvature tensors in a one-dimensional linear subspace of a manifold of the form T = T = p(M^{3}) generated by a kenmotsu manifold is shown to be zero.
Abstract: The purpose of this paper is to classify $\alpha$-para Kenmotsu manifolds $M^3$ such that the projection of the image of concircular curvature tensor $L$ in one-dimensional linear subspace of $T_{p}(M^{3})$ generated by $\xi_{p}$ is zero.
6 citations
01 Mar 2019
TL;DR: In this article, the authors studied the properties of N(k)-quasi Einstein manifolds and proved the existence of some classes of such manifolds by constructing physical and geometrical examples and showed that the characteristic vector field of the manifold is a unit parallel vector field as well as a Killing vector field.
Abstract: The object of the present paper is to study the properties of N(k)-quasi Einstein manifolds The existence of some classes of such manifolds are proved by constructing physical and geometrical examples It is also shown that the characteristic vector field of the manifold is a unit parallel vector field as well as Killing vector field
6 citations
Posted Content•
TL;DR: In this article, the authors investigate the geometry of foliations arising on conformally Einstein spaces (with Riemannian signature) where the non-degeneracy condition fails, which then allows them to characterize a general class of locally conformally-Einstein RiemANNian manifolds with degenerate Weyl tensors.
Abstract: The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M Listing, and in work by A R Gover and P Nurowski Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W We investigate the geometry of the foliations arising on conformally Einstein spaces (with Riemannian signature) where this condition fails, which then allows us to characterize a general class of locally conformally Einstein Riemannian manifolds with degenerate Weyl tensor
6 citations
Cites background or methods from "Riemannian Geometry of Contact and ..."
...…is called an associated metric to the contact structure (M, η) Sasakian structures arise in contact geometry by imposing a further integrability condition on the contact metric structure, which is analogous to the step from almost Kähler manifolds to Kähler in symplectic geometry (cf. [4] p. 71)....
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...Following [4], we define an almost contact metric structure as a set (M, η, g, φ) where, in addition to the contact structure, g is a Riemannian metric and φ a (1,1)-tensor satisfying φ 2 = −Id + η ⊗ ξ, such that g(φX, φY ) = g(X, Y ) − η(X)η(Y )....
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...Contact metric structures with a κ-nullity distribution (as well as more general distributions defined by curvature relations) have been studied quite a bit, especially in the case that the Reeb vector field ξ lies in the κ-nullity distribution (cf. [4] , p. 105 for some references)....
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