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On 3-dimensional (ε )-para Sasakian manifold · arXivDesk

1403.5090Mar 20, 2014

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

Punam Gupta

Differential Geometry

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 $\varphi$ - ${\cal T}$

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-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

% η

-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.

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