A solitonic study on para-Sasakian manifolds admitting semi-symmetric nonmetric connection

Abstract

In this paper we have introduced a new semi-symmetric nonmetric connection (briefly, SSNM-connection) and established its existence on para-Sasakian manifold. We obtain Riemannian curvature tensor, Ricci tensor, scalar curvature etc. with respect to the SSNM-connection and studied the properties of para-Sasakian manifold with the help of this connection. We also study $\eta $-Einstein soliton on para-Sasakian manifolds with respect to this connection and prove that a para-Sasakian manifold admitting $\eta $-Einstein soliton with respect to the SSNM-connection is a generalized $\eta$-Einstein manifold. Further, we investigate $\eta $-Einstein soliton on para-Sasakian manifolds satisfying $\overline{R}.\overline{S}=0,\overline{S}. \overline{R}=0$ and $\overline{R}.\overline{R}=0,$ where $\overline{R}$ and $\overline{S}$ are Riemannian curvature tensor and Ricci tensor with respect to the SSNM-connection, respectively. At last, some conclusions are made after observing all the results and an example of 3-dimensional para-Sasakian manifold admitting the SSNM-connection is given in which all the results can be verified easily.