Slant and Hemislant Submanifolds of a 3-Dimensional Indefinite Trans-Sasakian Manifold

Barnali Laha

doi:10.18052/www.scipress.com/bsmass.15.10

Abstract

In this paper we would like to establish some of the properties of slant and hemislant submanifoldsof an indefinite trans-Sasakian manifold. We have four sections in this paper. Section (1) isintroductory. In Section (2) we recall some necessary details of an indefinite trans-Sasakian manifold.In Section (3) we have obtained some interesting properties on a totally umbilical slant submanifoldsof an indefinite trans-Sasakian manifold. Finally, in Section (4), some results on integrability conditionsof the distributions of hemislant submanifolds of an indefinite trans-Sasakian manifold havebeen obtained.

Highlights

The study of slant submanifolds in complex spaces was initiated by B.Y.Chen as a natural generalization of both holomorphic and totally real submanifolds in ([1],[2])
Definition(2.1) : An n-dimensional Riemannian submanifold M of an indefinite trans-Sasakian manifold Mis called a contact CR-submanifold if i) ξ is tangent to M, ii) there exists on M a differentiable distribution D : x −→ Dx ⊂ Tx(M ), such that Dx is invariant under φ; i.e., φDx ⊂ Dx, for each x ∈ M and the orthogonal complementary distribution D⊥ : x −→ Dx⊥ ⊂ Tx⊥(M ) of the distribution D on M is totally real; i.e., φDx⊥ ⊂ Tx⊥(M ), where Tx(M ) and Tx⊥(M ) are the tangent space and the normal space of M at x respectively
If M is a slant submanifold of a 3−dimensional indefinite trans-Sasakian manifold, we can decompose tangent bundle T M of M as

Summary

Introduction

The study of slant submanifolds in complex spaces was initiated by B.Y.Chen as a natural generalization of both holomorphic and totally real submanifolds in ([1],[2]). Further modifications regarding semislant submanifolds were introduced by N.Papaghiuc [4]. These submanifolds are a generalized version of CR-submanifolds. J.L.Cabrerizo et al ([5],[6]) extended the study of semislant submanifolds of Kaehler manifold to the setting of Sasakian manifolds. Totally umbilical proper slant submanifold of a Kaehler manifold has been discussed in [7]. In 2011 Siraj Uddin et al studied totally umbilical proper slant and hemislant submanifolds of an LP-cosymplectic manifold. Our aim is to extend the study of slant and hemislant submanifolds of an indefinite trans-Sasakian manifold

Preliminaries

Slant submanifold

Hemislant submanifold