This study investigates the pseudo-projective curvature tensor within the framework of doubly and twisted warped product manifolds. It offers significant insights into the interaction between the pseudo-projective curvature tensor and both the base and fiber manifolds. The research highlights key geometric characteristics of the base and fiber manifolds as influenced by the pseudo-projective curvature tensor in these structures. Additionally, the paper extends its analysis to examine the behavior of the pseudo-projective curvature tensor in the context of generalized doubly and twisted generalized Robertson-Walker space-times.

Recently, B.-Y. Chen discovered a technique to find the relation between second fundamental form and the warping function of warped product submanifolds. In this paper, we extend our further study of [24] by giving non-trivial examples of warped product pointwise hemi-slant submanifolds. Finally, we establish a sharp estimation for the squared norm of the second fundamental form ||h||2 in terms of the warping function f. The equality case is also investigated.

In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen’s inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped products are also proved in this paper.

In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products. We establish a lower bound relationship between the squared norm of the second fundamental form and the warping function. The equality case of the inequality is investigated and some special cases of derived inequality are given. Furthermore, we provide non-trivial examples of such submanifolds.

This research aims to prove a sharp relationship between statistical submersions and doubly minimal immersions. We consider a non-trivial statistical submersion on a statistical manifold with isometric fibers. Then, we investigate that it cannot be isometrically immersed as a doubly minimal manifold into any statistical manifold of non-positive sectional curvature using a submersion invariant for dual connections and additional conditions.