On $3$-dimensional $\left(\varepsilon \right)$-para Sasakian manifold

Abstract

The purpose of the present paper is to study the globally and locally $\varphi $-${\cal T}$-symmetric $\left( \varepsilon \right) $-para Sasakian manifold in dimension $3$. The globally $\varphi $-$ {\cal T}$-symmetric $3$-dimensional $\left( \varepsilon \right) $-para Sasakian manifold is either Einstein manifold or has a constant scalar curvature. The necessary and sufficient condition for Einstein manifold to be globally $\varphi $-${\cal T}$ -symmetric is given. A $3$-dimensional $% \left( \varepsilon \right) $ -para Sasakian manifold is locally $\varphi $-$ {\cal T}$-symmetric if and only if the scalar curvature $r$ is constant. A $3 $-dimensional $\left( \varepsilon \right) $-para Sasakian manifold with $% \eta $-parallel Ricci tensor is locally $\varphi $-${\cal T}$-symmetric. In the last, an example of $3$-dimensional locally $\varphi $-${\cal T}$-symmetric $\left( \varepsilon \right) $-para Sasakian manifold is given.