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.

In this research note, we investigated the characteristics of perfect fluid spacetime when coupled with the hyperbolic Ricci soliton. We additionally interacted with the perfect fluid spacetime, with a $\phi (\mathcal{Q})$-vector field and a bi-conformal vector field that admits the hyperbolic Ricci solitons. Furthermore, we analyze the gradient hyperbolic Ricci soliton in perfect fluid spacetime, employing a scalar concircular field, and discuss about the gradient hyperbolic Ricci soliton's rate of change. In the end, we determined the energy conditions for perfect fluid spacetime in terms of gradient hyperbolic Ricci soliton with a scalar concircular field.

In this paper, we introduce the notion of a bi-slant lightlike submanifold of a golden semi-Riemannian manifold. We provide two examples. We give some characterizations about the geometry of such submanifolds.

In this article, we derive a Schur-type Inequality in terms of the gradient $r$-Almost Newton-Ricci-Yamabe soliton in $(k,\mu )$-contact metric manifolds. We discuss the triviality for the compact gradient $r$-Almost Newton-Ricci-Yamabe soliton in $(k,\mu )$-Contact metric manifolds. In the end, we deduce a Schur-type inequality for the gradient $r$-Almost Newton-Yamabe soliton in $(k,\mu )$-contact metric manifolds, static Riemannian manifolds, and normal homogeneous compact Riemannian manifolds coupled with a projected Casimir operator.