Some Curvature Conditions on a Para-Sasakian Manifold with Canonical Paracontact Connection

Bilal Eftal Acet,Erol Kılıç,Selcen Yüksel Perktaş

doi:10.1155/2012/395462

Abstract

We study canonical paracontact connection on a para-Sasakian manifold. We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is anη-Einstein manifold. We also investigate some properties of curvature tensor, conformal curvature tensor,W2-curvature tensor, concircular curvature tensor, projective curvature tensor, and pseudo-projective curvature tensor with respect to canonical paracontact connection on a para-Sasakian manifold. It is shown that a concircularly flat para-Sasakian manifold with respect to canonical paracontact connection is of constant scalar curvature. We give some characterizations for pseudo-projectively flat para-Sasakian manifolds.

Highlights

In 1976, Sato 1 introduced the almost paracontact structure φ, ξ, η satisfying φ2 I − η ⊗ ξ and η ξ 1 on a differentiable manifold
We study canonical paracontact connection on a para-Sasakian manifold
We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is an η-Einstein manifold

Summary

Introduction

In 1976, Sato 1 introduced the almost paracontact structure φ, ξ, η satisfying φ2 I − η ⊗ ξ and η ξ 1 on a differentiable manifold. In this study we define a canonical paracontact connection on a para-Sasakian manifold which seems to be the paracontact analogue of the generalized Tanaka-Webster connection. It is proved that if the condition Z ξ, X · S 0 holds on a paraSasakian manifold with respect to canonical paracontact connection, the scalar curvature is constant. P P 0 holds, is either of constant scalar curvature or an η-Einstein manifold

Preliminaries

Curvature Tensor

Conformal Curvature Tensor

W2-Curvature Tensor

Concircularly Flat Para-Sasakian Manifold

Pseudo-projectively Flat Para-Sasakian Manifold