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.

The main purpose of this paper is to study the properties of $\mathcal{P}\mathcal{R}$-semi-invariant submanifold of para-Kenmotsu manifold. We obtain the integrability conditions for the invariant distribution and anti-invariant distribution. We obtain some existence and non-existence results of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds. We provide some necessary and sufficient conditions for $\mathcal{P}\mathcal{R}$-semi-invariant submanifold to be a $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold in para-Kenmotsu manifold. We also derive some sharp inequalities for $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold in para-Kenmotsu manifolds.

Our investigation involved sequentially warped product manifolds that contained gradient-normalized Ricci-harmonic solitons. We presented the primary connections for a gradient-normalized Ricci-harmonic soliton on sequential warped product manifolds. In practical applications, our research investigated gradient-normalized Ricci-harmonic solitons for sequential generalized Robertson-Walker spacetimes and sequential standard static space-times. Our finding generalized all results proven in ^[26].