A study of the relationship between pseudo-umbilical totally real submanifolds and minimal totally real submanifolds in complex space forms is presented in this paper. The paper studies totally real submanifolds in complex space forms. The moving-frame method and the DDVV inequality (a conjecture for the Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken) are used to obtain some rigidity theorems and an integral inequality, improving the associated results.

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.