Contact CR-submanifolds Research Articles

Submanifolds of some indefinite contact and paracontact manifolds

The purpose of the present paper is to study some types of submanifolds of indefinite contact and paracontact manifolds and a few properties of contact CR-submanifolds of an indefinite Sasakian manifold, indefinite Kenmotsu manifold, indefinite trans-Sasakian manifold and ε-paracontact Sasakian manifold.

A note on warped product submanifolds of Kenmotsu manifolds

Abstract Warped product manifolds are known to have applications in Physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [HONG, S. T.: Warped products and black holes, Nuovo Cimento Soc. Ital. Fis. B 120 (2005), 1227–1234]). The studies on warped product manifolds with extrinsic geometric point of view are intensified after B. Y. Chen’s work on CR-warped product submanifolds of Kaehler manifolds. Later on, similar studies are carried out in the setting of Sasakian manifolds by Hasegawa and Mihai. As Kenmotsu manifolds are themselves warped product spaces, it is interesting to investigate warped product submanifolds of Kenmotsu manifolds. In the present note a larger class of warped product submanifolds than the class of contact CR-warped product submanifolds is considered. More precisely the existence of warped product submanifolds of a Kenmotsu manifold with one of the factors an invariant submanifold is ensured, an example of such submanifolds is provided and a characterization for a contact CR-submanifold to be a contact CR-warped product submanifold is established.

SCALAR CURVATURE OF CONTACT CR-SUBMANIFOLDS IN AN ODD-DIMENSIONAL UNIT SPHERE

In this paper we derive an integral formula on an (n + 1)-dimensional, compact, minimal contact CR-submanifold M of (n - 1) contact CR-dimension immersed in a unit (2m+1)-sphere <TEX>$S^{2m+1}$</TEX>. Using this integral formula, we give a sufficient condition concerning with the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.

A note on doubly warped product contact C R-submanifolds in trans-Sasakian manifolds

Warped product CR-submanifolds in Kaehlerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry, was developed, namely warped product contact CR-submanifolds in Sasakian manifolds. In this note we proved that there exists no proper doubly warped product contact CR-submanifolds in trans-Sasakian manifolds.

CERTAIN CONTACT CR-SUBMANIFOLDS OF AN ODD-DIMENSIONAL UNIT SPHERE

We study an (n+1)(n ≥ 3)-dimensional contact CR-submanifold of (n−1) contact CR-dimension in a (2m+1)-unit sphere S2m+1, and to determine such submanifolds under conditions concerning the second fundamental form and the induced almost contact structure.

Warped product contact CR-submanifolds of trans-Sasakian manifolds

B.Y. Chen [4] showed that there exists no proper warped CR-submanifolds N_ x ? NT of a Kaehler manifold and obtained many results on CR-warped products NT x ? N_. Contact CR-warped product submanifolds in Sasakian manifold were studied by I. Hasegawa and I. Mihai [6]. In this paper we have investigated the existence of contact CR-warped product submanifolds in more general setting of trans-Sasakian manifolds. .

SECTIONAL CURVATURE OF CONTACT C R-SUBMANIFOLDS OF AN ODD-DIMENSIONAL UNIT SPHERE

In this paper we study (n + 1)-dimensional compact contact CR-submanifolds of (n - 1) contact CR-dimension immersed in an odd-dimensional unit sphere <TEX>$S^{2m+1}$</TEX>. Especially we provide necessary conditions in order for such a sub manifold to be the generalized Clifford surface <TEX>$$S^{2n_1+1}(((2n_1+1)/(n+1))^{\frac{1}{2}})\;{\times}\;S^{2n_2+1}(((2n_2+1)/(n+1)^{\frac{1}{2}})$$</TEX> for some portion (n1, n2) of (n - 1)/2 in terms with sectional curvature.

Contact CR-Submanifolds of an Odd-Dimensional Unit Sphere

We study an (n+1)(n≥ 3)-dimensional contact CR-submanifold of (n−1) contact CR-dimension in a (2m+1)-unit sphere S2m+1, and especially determine such submanifolds under the equality conditions appearing in (3.12). We also provide a sufficient condition in order for such a compact submanifold to be the model space $$S^{2n_1+1}(1/\sqrt2) \times S^{2n_2+1}(1/\sqrt2)$$ given in the last Section 4.

(n + 1)-DIMENSIONAL, CONTACT CR-SUBMANIFOLDS OF (n - 1) CONTACT CR-DIMENSION IN A SASAKIAN SPACE FORM

In this paper. We Study (n + 1)-dimensional Contact CR-submanifolds of (n - 1) contact CR-dimension immersed in a Sasakian space form M<TEX>$\^$</TEX>2m+1/(c) (2m=n+p, p>0), and especially determine such submanifolds under additional condition concerning with shape operator.

Contact CR-submanifolds in Sasakian manifolds -- a foliated approach

Contact CR-submanifolds in Sasakian manifolds -- a foliated approach

Normal contact CR-submanifolds of a quasi-Sasakian manifold

Normal contact CR-submanifolds of a quasi-Sasakian manifold

Contact CR-submanifolds with parallel mean curvature vector of a Sasakian space form

Contact CR-submanifolds with parallel mean curvature vector of a Sasakian space form

Space time manifolds and contact structures

A new class of contact manifolds (carring a global non‐vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4‐dimensional spacetime of general relativity as a contact CR‐submanifold.

Automorphisms and a reduction theorem in a Sasakian space form E2m+1(−3)

We construct definitely the automorphism group of a Sasakian space form ¯M=E2m+1(−3) and study the existence of a totally geodesic invariant submanifold of ¯M tangent to a given invariant subspace in the tangent space of ¯M. We also study the Frenet curves in ¯M under a totally contact geodesic immersion of a contact CR-submanifold into ¯M. The purpose of this paper is to prove a reduction theorem of the codimension for a totally contact geodesic, contact CR-submanifold of ¯M.

Contact CR submanifolds

Let \(\bar M\) be a (2m+1)-dimensional Sasakian manifold with structure tensors (φ,ξ,η,g). We consider a Riemannian manifold isometrically immersed in \(\bar M\) with induced metric tensor field g.