Contact geometry and CR-structures on spheres (original) (raw)

Five-Dimensional Contact CR-Submanifolds in S 7 ( 1 )

Mathematics

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure (φ,ξ,η), we study its five-dimensional contact CR-submanifolds, which are the analogue of CR-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M, the tangent bundle of contact CR-submanifold M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant and E(M) is anti-invariant with respect to φ. On this occasion we obtain a complete classification of five-dimensional proper contact CR-submanifolds in S7(1) whose second fundamental form restricted to H(M) and E(M) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.

New aspects on CR-structures of codimension 2 on

2016

Abstract. We introduce a torsion free linear connection on a hypersurface in a Sasakian manifold on which we have defined in natural way a CR-structure of CR-codimension 2. We study the curvature properties of this connection and we give some interesting examples.

Contact CR-warped product submanifolds of (LCS)_n-manifolds

2017

The present paper deals with a study of doubly warped product contact CR-submanifolds of (LCS)n-manifolds and warped product contact CR-submanifolds of (LCS)_n-manifolds. It is shown that there exists no doubly warped product contact CR-submanifolds of (LCS)_n-manifolds. However we obtain some results for the existence or non-existence of warped product contact CR-submanifolds of (LCS)_n-manifolds and the existence is also ensured by an interesting example.

On CR-submanifolds of the six-dimensional sphere

International Journal of Mathematics and Mathematical Sciences, 1995

We consider properCR-submanifolds of the six-dimensional sphereS6. We prove thatS6does not admit compact properCR-submanifolds with non-negative sectional curvature and integrable holomorphic distribution.

Classification of contact structures associated with the CR-structure of the complex indicatrix

Analele Universitatii "Ovidius" Constanta - Seria Matematica

By regarding the complex indicatrix as an embedded CR-hypersurface of the holomorphic tangent bundle in a fixed point, we analyze some aspects of the relations between its CR structure and the considered contact structure. Moreover, using the classification of the almost contact metric structures associated with a strongly pseudo-convex CR-structure, of D. Chinea and C. Gonzales, we determine the classes corresponding to the natural contact structure of the complex indicatrix and the new structures obtained under a gauge transformation.

On contact CR-submanifolds of a cosymplectic manifold

Filomat

In this paper, we study the differential geometry of contact CR-submanifolds of a cosymplectic manifold. Necessary and sufficient conditions are given for a submanifold to be a contact CR-submanifold in cosymplectic manifolds and cosymplectic space forms. Finally, the induced structures on submanifolds are investigated, these structures are categorized and we discuss these results.

Prescribing the Webster scalar curvature on CR spheres

Comptes Rendus Mathematique, 2011

In this Note, we give two new perturbative results for prescribing the Webster scalar curvature on the (2n + 1)-dimensional sphere endowed with its standard CR structure. The first result generalizes the one obtained by A. Malchiodi and F. Uguzzoni (2002) in [9].