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.

This paper introduces twisted sequential warped products. This new class of warped products extends solutions to Einstein’s field equations. We derive explicit formulas for covariant derivatives, curvature tensors, Ricci curvature, and scalar curvature. We characterize Einstein conditions and analyze geodesics. We also study conformal vector fields, including Killing and concircular types. Furthermore, we examine gradient almost Ricci solitons and the concircular curvature tensor. Explicit examples of such manifolds are presented, and applications to cosmology, black hole physics, and quantum field theory are discussed. These findings deepen the understanding of warped product geometry in general relativity. They also provide a foundation for further research.