LP-Sasakian Manifolds Equipped with Zamkovoy Connection and Conharmonic Curvature Tensor

Ashoke Das,Abhijit Mandal

doi:10.22342/jims.27.2.960.137-149

Abstract

In this paper we have proved some results on conharmonically flat, quasi conharmonically flat and φ-conharmonically flat LP-Sasakian manifolds with respect to Zamkovoy connection. Also, we study generalized conharmonic φ-recurrent LP-Sasakian manifolds with respect to Zamkovoy connection. Moreover, we study LP-Sasakian manifolds satisfying K*(ξ,U)∘R*=0, where K* denotes conharmonic curvature tensor and R* denotes Riemannian curvature tensor with respect to Zamkovoy connection.

Highlights

The generalized recurrent manifolds was introduced by Dubey [8] and it was studied by De and Guha et al [6]
An n−dimensional LP-Sasakian manifold M is said to be generalized conharmonic φ−recurrent with respect to Zamkovoy connection if φ2 (∇∗W K∗) (X, Y ) Z = A (W ) K (X, Y ) Z +B (W ) [g (Y, Z) X − g (X, Z) Y ], for all X, Y, Z, W ∈ χ (M ), where A and B are 1−forms and B is non vanishing such that A (W ) = g (W, ρ1), B (W ) = g (W, ρ2) and ρ1, ρ2 are vector fields associated with 1−forms A and B, respectively
The Zamkovoy connection on an n−dimensional LP-Sasakian manifold is a non-metric linear connection with torsion tensor given by equation (20)

Summary

Introduction

The conharmonic curvature tensor(K∗) with respect to Zamkovoy connection is given by. An n−dimensional LP-Sasakian manifold M is said to be generalized conharmonic φ−recurrent with respect to Zamkovoy connection if φ2 (∇∗W K∗) (X, Y ) Z = A (W ) K (X, Y ) Z +B (W ) [g (Y, Z) X − g (X, Z) Y ] , for all X, Y, Z, W ∈ χ (M ) , where A and B are 1−forms and B is non vanishing such that A (W ) = g (W, ρ1) , B (W ) = g (W, ρ2) and ρ1, ρ2 are vector fields associated with 1−forms A and B, respectively. In section (3), we have obtained Riemannian curvature tensor R∗, Ricci tensor S∗, scalar curvature r∗ with respect to Zamkovoy connection in LP-Sasakian manifold.

Preliminaries

Some properties of LP-Sasakian manifold with respect to Zamkovoy connection