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.

In this work, using optimization procedures on Riemannian submanifolds, we obtained two different inequalities on the generalized normalized $\delta$-Casorati curvatures of warped product submanifolds in complex space forms. We also quantified the conditions under which these inequalities become equalities, providing more insight into their geometric consequences. Further, we described new findings in the form of harmonic functions and Hessian functions, which offer a more general view of the interplay between curvature and analyticity.

This study explored specific inequalities related to the scalar and Ricci curvatures of slant submersions in Kenmotsu space forms. We derived important geometric bounds and systematically investigated the conditions under which these bounds converge to equality. These findings enlarge the current setup of curvature inequalities and offer new findings on the geometric properties of slant submersions of contact structures.

In this paper, we established the sharp Chen-type inequalities for $QR$-submanifolds immersed in quaternionic space forms endowed with a quarter-symmetric metric connection (QSMC). By extending Chen's $\delta$-invariant framework to both the invariant and anti-invariant distributions of a $QR$-submanifold, we obtained optimal upper bounds for the invariants $\delta (\mathcal{D})$ and $\delta ({\mathcal{D}}^{\perp })$. The resulting inequalities explicitly capture the influence of the quarter-symmetric metric connection through its structural parameters and associated tensor fields, thereby providing a unified generalization of the classical Levi-Civita and semi-symmetric settings. Furthermore, we completely characterized the equality cases, showing that the bounds are attained precisely when the submanifold is mixed geodesic, and the invariant distribution is totally umbilical under specific constraints on the second fundamental form. As direct consequences, previously known inequalities for semi-symmetric metric and nonmetric connections are recovered. These results furnish a new rigidity phenomena for extremal $QR$-submanifolds and deepen the understanding of curvature invariants in quaternionic geometry.