On contact CR-submanifolds of a cosymplectic manifold

New Trends in Mathematical Science

This paper is concerned with the study of the contact pseudo-slant submanifolds of a cosymplectic manifold. We derive the integrability conditions of involved distributions in the definition of a pseudo-slant submanifold. The notion contact parallel and contact pseudo-slant product is defined and the necessary and sufficient conditions for a submanifold to be contact parallel and contact pseudo-slant product are given. Also, an non-trivial example is used to demonstrate that the method presented in this paper is effective.

International Journal of Mathematics and Mathematical Sciences, 1983

Recently, K.Yano and M.Kon [5] have introduced the notion of a contactCR-submanifold of a Sasakian manifold which is closely similar to the one of aCR-submanifold of a Kaehlerian manifold defined by A. Bejancu [1].In this paper, we shall obtain some fundamental properties of contactCR-submanifolds of a Sasakian manifold. Next, we shall calculate the length of the second fundamental form of a contactCR-product of a Sasakian space form (THEOREM 7.4). At last, we shall prove that a totally umbilical contactCR-submanifold satisfying certain conditions is totally geodesic in the ambient manifold (THEOREM 8.1).

Differential Geometry-Dynamical …, 2010

In this paper, we prove that there do not exist contact totally umbilical proper slant submanifolds of cosymplectic manifolds.

Journal of Differential Geometry

We prove the existence of contact submanifolds realizing the Poincaré dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact category to Donaldson's construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiate sense and that satisfy a transversality with estimates property directly in the contact category. The description of the obtained contact submanifolds allows us to prove an extension of the Lefschetz hyperplane theorem which completes their topological characterization.

Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics

In the present paper, we study totally umbilical submanifolds of cosymplectic manifolds. We obtain a result on the classification of totally umbilical contact CR-submanifolds of a cosymplectic manifold.

Kodai Mathematical Journal, 1981

The purpose of the present paper is to study the structure of almost cosymplectic manifolds. §2 presents the basic definitions and some preliminary properties of an almost cosymplectic structure. Examples of such structures are given in § 3. In § 4 we state many important curvature identities for almost cosymplectic manifolds. In § 5 we give certain sufficient conditions for an almost contact metncstructure to be almost cosymplectic. Basing on the identities from §4 we prove in §6 that almost cosymplectic manifolds of non-zero constant sectional curvature do not exist in dimensions greater than three. However it is known (cf. [2]) that such manifolds of zero sectional curvature (i. e. locally flat) exist and they are cosymplectic. Moreover we give certain restrictions on the scalar curvature of almost cosymplectic manifolds which are conformally flat or of constant ^-sectional curvature. All manifolds considered in this paper are assumed to be connected and of class C°°. All tensor fields, including differential forms, are of class C°°. The notation and terminology will be the same as that employed in [2]. § 2. Preliminaries. Let (M, φ, ξ, 77, g) be a (2n+l)-dimensional almost contact metric manifold, that is, Mis a differentiate manifold and {φ, ξ, η, g) an almost contact metric structure on M, formed by tensor fields φ, ξ, η of type (1, 1), (1, 0), (0, 1), respectively, and a Riemannian metric g such that φξ=O, η°φ=0, η(X)=g(X, ξ), g(φX, φY)=g(X, Y)-η(X)η{Y). On such manifold we may always define a 2-form Φ by Φ(X, Y)~g(φX, Y). (Λf, φ, ξ, 7], g) is said to be an almost cosymplectic manifold (cf. [2]) if the forms Φ and Ύ] are closed, i.e. dΦ=0 and dη^O, where d is the operator of exterior differentiation. In particular, if the almost contact structure of an almost cosymplectic manifold is normal, then it is said to be a cosymplectic manifold (cf. [1]).

In the present paper, we study totally umbilical submanifolds of cosym-plectic manifolds. We obtain a result on the classification of totally umbilical contact CR-submanifolds of a cosymplectic manifold.

2018

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

Vietnam Journal of Mathematics, 2013

We introduce GCR-lightlike submanifolds of indefinite Cosymplectic manifolds and give an example. We also study the existence of this class in indefinite Cosymplectic space form. Then we prove that there does not exist a totally contact umbilical GCR-lightlike submanifold of an indefinite Cosymplectic manifold other than totally contact geodesic GCR-lightlike submanifold and moreover it is a totally geodesic GCR-lightlike submanifold.

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