Contact manifolds and generalized complex structures

Comptes Rendus Mathematique, 2016

In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators [9], it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.

Mediterranean Journal of Mathematics, 2016

The notions of a twistor space of a contact manifold and a contact connection on such a manifold have been introduced by L. Vezzoni as extensions of the corresponding notions in the case of a symplectic manifold. Given a contact connection on a contact manifold one can define an almost CR-structure on its twistor space and Vezzoni has found the integrability condition for this structure. In the present paper it is observed that the CR-structure is induced by an almost contact metric structure. The main goal of the paper is to obtain necessary and sufficient conditions for normality of this structure in terms of the curvature of the given contact connection. Illustrating examples are discussed at the end of the paper.

Kodai Mathematical Journal, 1981

We determine an orthogonal decomposition of the vector space of some curvature tensors on a co-Hermitian real vector space, in irreducible components with respect to the natural induced representation of c U(n)xl.

Filomat, 2015

We consider a pair of smooth manifolds, which are the counterparts in the even-dimensional and odd-dimensional cases. They are separately an almost complex manifold with Norden metric and an almost contact manifolds with B-metric, respectively. They can be combined as the so-called almost contact complex Riemannian manifold. This paper is a survey with additions of results on differential geometry of canonical-type connections (i.e. metric connections with torsion satisfying a certain algebraic identity) on the considered manifolds.

Inventiones mathematicae, 2000

The present work is concerned with the study of complex projective manifolds X which carry an additional complex contact structure. In the first part of the paper we show that if K X is not nef, then either X is Fano and b 2 (X) = 1, or X is of the form P(T Y ), where Y is a projective manifold. In the second part of the paper we consider contact manifolds where K X is nef or, more generally, complex projective manifolds where the bundle of holomorphic 1-forms contains a nef sub-bundle of rank 1.

Journal of Geometry and Physics, 2016

A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian manifold from a holomorphic complex Riemannian manifold is presented and called an S 1 -solvable extension.

Georgian Mathematical Journal

We study the conformal curvature tensor and the contact conformal curvature tensor in Sasakian and/or K-contact manifolds. We find a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat. We also find some necessary conditions for a K-contact manifold to be ϕ-contact conformally flat. Then we give a structure theorem for ϕ-contact conformally flat Sasakian manifolds. It is also proved that a Sasakian manifold cannot be ξ-contact conformally flat.

Journal Fur Die Reine Und Angewandte Mathematik, 2003

We classify locally homogeneous quasi-Sasakian manifolds in dimension five that admit a parallel spinor $\psi$ of algebraic type $F \cdot \psi = 0$ with respect to the unique connection $\nabla$ preserving the quasi-Sasakian structure and with totally skew-symmetric torsion. We introduce a certain conformal transformation of almost contact metric manifolds and discuss a link between them and the dilation function in 5-dimensional string theory. We find natural conditions implying conformal invariances of parallel spinors. We present topological obstructions to the existence of parallel spinors in the compact case.

GANIT: Journal of Bangladesh Mathematical Society, 2019

In this paper, we discuss about almost complex structures and complex structures on Riemannian manifolds, symplectic manifolds and contact manifolds. We have also shown a special comparison between complex symplectic geometry and complex contact geometry. Also, the existence of a complex submanifold of n-dimensional complex manifold which intersects a real submanifold GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 119-126

Geometriae Dedicata, 2010

We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun and Salamon on Fano contact manifolds but under a symmetry assumption instead of a curvature condition.