On $(\varepsilon)$-para Sasakian 3-manifolds

TL;DR

This paper characterizes 3-dimensional $( abla)$-para Sasakian manifolds, establishing conditions for them to be indefinite space forms, and explores their curvature properties and symmetries.

Contribution

It provides necessary and sufficient conditions for $( abla)$-para Sasakian 3-manifolds to be indefinite space forms and analyzes their curvature and symmetry properties.

Findings

$( abla)$-para Sasakian 3-manifolds are pseudosymmetric.

Ricci-semi-symmetric manifolds are indefinite space forms.

Scalar curvature is constant when Ricci tensor is $ abla$-parallel.

Abstract

In this paper we study the 3-dimensional $(\varepsilon) $-para Sasakian manifolds. We obtain an necessary and sufficient condition for an $(\varepsilon ) $-para Sasakian 3 -manifold to be an indefinite space form. We show that a Ricci-semi-symmetric $(\varepsilon) $-para Sasakian 3 -manifold is an indefinite space form. We investigate the necessary and sufficient condition for an $(\varepsilon) $-para Sasakian 3 -manifold to be locally $\varphi $-symmetric. It is proved that in an $ (\varepsilon) $-para Sasakian 3-manifold with $\eta $ -parallel Ricci tensor the scalar curvature is constant. It is also shown that every $(\varepsilon) $-para Sasakian 3-manifolds is pseudosymmetric in the sense of R. Deszcz.