Lorentzian $\alpha$-Sasakian Manifold with $\mathbf{W}_{3}$-Curvature Tensor

In the present paper, we investigate Lorentzian $\alpha $-Sasakian manifolds equipped with a semi-symmetric non-metric (SSNM) connection. First, we establish the existence and uniqueness of the SSNM-connection and derive several fundamental curvature properties associated with this connection. We then study the behavior of the $W_{3}$-curvature tensor on such manifolds and obtain characterizations of $\xi $-$\overline{W}_{3}$ -flat, $\phi $-$\overline{W}_{3}$-flat, and $\phi $-$\overline{W}_{3}$ -semi-symmetric Lorentzian $\alpha $-Sasakian manifolds. It is shown that under suitable curvature conditions, namely $\phi $-$\overline{W}_{3}$ -flatness and $\overline{W}_{3}.\overline{R}=0$, the manifold becomes a generalized $\eta $-Einstein manifold. Furthermore, scalar curvature relations are obtained for manifolds satisfying $\xi $-$\overline{W}_{3}$ -flatness, $\overline{W}_{3}.\overline{S}=0,$ $\overline{S}.\overline{W} _{3}=0$, and $\phi $-$\overline{W}_{3}$-semi-symmetry. Finally, a non-trivial $3$-dimensional example of a Lorentzian $\alpha $-Sasakian manifold admitting an SSNM-connection is constructed to verify the theoretical results. Here, $\overline{W}_{3},\overline{R}$ and $\overline{S} $ denote the $W_{3}$-curvature tensor, Riemannian curvature tensor and Ricci curvature tensor with respect to the SSNM-connection.