Some notes on $LP$-Sasakian Manifolds with Generalized Symmetric Metric Connection

The present study initially identify the generalized symmetric connections of type $(\alpha ,\beta )$, which can be regarded as more generalized forms of quarter and semi-symmetric connections. The quarter and semi-symmetric connections are obtained respectively when $(\alpha ,\beta )=(1,0)$ and $(\alpha ,\beta )=(0,1)$. Taking that into account, a new generalized symmetric metric connection is attained on Lorentzian para-Sasakian manifolds. In compliance with this connection, some results are obtained through calculation of tensors belonging to Lorentzian para-Sasakian manifold involving curvature tensor, Ricci tensor and Ricci semi-symmetric manifolds. Finally, we consider $CR$-submanifolds admitting a generalized symmetric metric connection and prove many interesting results.