Indefinite trans-Sasakian manifold with semi-symmetric metric connection

Abstract

The objective of the present paper is to study of indefinite trans-Sasakian manifold with a semi-symmetric metric connection. We have found the relations between curvature tensors, Ricci curvature tensors and scalar curvature of indefinite trans-Sasakian manifolds with semi-symmetric metric connection and with metric connection. Also, we have proved some results on quasi-projectively flat and $\varphi -$projectively flat manifolds with respect to semi-symmetric metric connection. It is shown that the manifold satisfying $\overset{\_}{R.}\overset{\_}{S}$ $=0$ is an $\eta -$Einstein manifold if $\alpha =0$ and $\beta =constant.$ It is also proved that the manifold satisfying $\overset{\_}{P}.\overset{\_}{S}=0$ is an $\eta -$ Einstein manifold if $\alpha =0$ and $\beta =constant.$ Finally, we have obtained the conditions for the manifold with semi-symmetric metric connection to be conformally flat and $\xi -$conformally flat.