GRADIENT RICCI SOLITONS IN δ-LORENTZIAN TRANS-SASAKIAN MANIFOLDS WITH SEMI-SYMMETRIC METRIC CONNECTION

Abstract

The aim of the present research paper is to study the $\delta$-Lorentzian Trans Sasakian manifolds endowed semi-symmetric metric connections addmitting the gradient Ricci Solitons, $\eta$-Ricci Solitons and Ricci Solitions. Initialy, it is shown that the $\delta$-Lorentzian trans Sasakian manifolds with a semi-symmetric-metric connection. We have found the expressions for curvature tensors, Ricci curvature tensors and scalar curvature of the $\delta$-Lorentzian trans Sasakian manifolds with a semi-symmetric-metric and metric connection. Also, we have discussed some results on quasi-projectively flat and $\phi$-projectively flat manifolds endowed with a semi-symmetric-metric connection. It shows that the manifold satisfying $\bar{R}.\bar{S}=0$, $\bar{P},\bar{S}=0$. Moreover, we have obtained the conditions for the $\delta$-Lorentzian Trans Sasakian manifolds with a semi-symmetric-metric connection to be conformally flat and $\xi$-conformally flat.