Lorentzian α-Sasakian Manifold with W₃-Curvature Tensor

In the present paper, we investigate Lorentzian α-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 W₃-curvature tensor on such manifolds and obtain characterizations of ξ-W̄₃ -flat, φ-W̄₃-flat, and φ-W̄₃ -semi-symmetric Lorentzian α-Sasakian manifolds. It is shown that under suitable curvature conditions, namely φ-W̄₃ -flatness and W̄₃.R̄=0, the manifold becomes a generalized η-Einstein manifold. Furthermore, scalar curvature relations are obtained for manifolds satisfying ξ-W̄₃ -flatness, W̄₃.S̄=0, S̄.W̄ ₃=0, and φ-W̄₃-semi-symmetry. Finally, a non-trivial $3$-dimensional example of a Lorentzian α-Sasakian manifold admitting an SSNM-connection is constructed to verify the theoretical results. Here, W̄₃,R̄ and S̄ denote the W₃-curvature tensor, Riemannian curvature tensor and Ricci curvature tensor with respect to the SSNM-connection.