Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products $\mathbb{R}{\times }_{\mathfrak{f}}{N}_2$ and $N_1{\times }_{\mathfrak{f}}\mathbb{R}$ . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

The goal of the present research paper is to study how a spacetime manifold evolves when thermal flux, thermal energy density and thermal stress are involved; such spacetime is called a thermodynamical fluid spacetime (TFS). We deal with some geometrical characteristics of $TFS$ and obtain the value of cosmological constant $\mathsf{\Lambda }$. The next step is to demonstrate that a relativistic $TFS$ is a generalized Ricci recurrent $TFS$. Moreover, we use $TFS$ with thermodynamic matter tensors of Codazzi type and Ricci cyclic type. In addition, we discover the solitonic significance of $TFS$ in terms of the Ricci metric (i.e., Ricci soliton $RS$).

In the present paper, weinvestigate some properties of the distributions involved in the definition of a CR-statistical submanifold. The characterization of a CR-product in holomorphic statistical manifolds is given. By using an optimization technique, we establish a relationship between the Ricci curvature and the squared norm of the mean curvature of any submanifold in the same ambient space. The equality case is also discussed here. This paper finishes with some related examples.

Kenmotsu geometry is a valuable part of contact geometry with nice applications in other fields such as theoretical physics. In this article, we study the statistical counterpart of a Kenmotsu manifold, that is, Kenmotsu statistical manifold with some related examples. We investigate some statistical curvature properties of Kenmotsu statistical manifolds. It has been shown that a Kenmotsu statistical manifold is not a Ricci-flat statistical manifold by constructing a counter-example. Finally, we prove a very well-known Chen-Ricci inequality for statistical submanifolds in Kenmotsu statistical manifolds of constant ?-sectional curvature by adopting optimization techniques on submanifolds. This article ends with some concluding remarks.

In this paper, we use T Oprea's optimization method to establish some optimal Casorati inequalities, which involve the normalized scalar curvature for bi-slant submanifolds of generalized Sasakian space forms. In the continuation, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical.