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

Abstract

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