Geometric aspects of weakly symmetric and almost pseudo symmetric $K$-contact manifolds admitting a non-symmetric non-metric connection

In this paper, we explore the conditions for several geometric structures-specifically, weakly symmetric, weakly Ricci symmetric, almost pseudo symmetric, and almost pseudo Ricci symmetric–on a $K$-contact manifold equipped with a non-symmetric non-metric connection. We present key theoretical results that characterize these structures in the context of such a connection. To illustrate the applicability of our findings, we construct an explicit example of a 3-dimensional $K$-contact manifold with a non-symmetric non-metric connection. This example not only confirms the validity of the derived conditions but also offers a concrete model for further investigation in the field. Our results extend the understanding of geometric structures on $K$-contact manifolds beyond the classical framework of the Levi-Civita connection, paving the way for new directions in differential geometry and mathematical physics.