In the present paper, we investigate the nature of Ricci-Yamabe soliton on an imperfect fluid generalized Robertson-Walker spacetime with a torse-forming vector field . Furthermore, if the potential vector field of the Ricci-Yamabe soliton is of the gradient type, the Laplace-Poisson equation is derived. Also, we explore the harmonic aspects of -Ricci-Yamabe soliton on an imperfect fluid spacetime with a harmonic potential function . Finally, we examine necessary and sufficient conditions for a -form , which is the -dual of the vector field on imperfect fluid spacetime to be a solution of the Schrödinger-Ricci equation.

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for $\delta (2,2)$-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end.

In [3], it was shown that there are no warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold of a warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function and show that there exists a class of non-trivial warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold is pointwise slant by giving some examples. We present a characterization theorem and establish a sharp relationship between the squared norm of the second fundamental form and the warping function in terms of the slant function for such warped product submanifolds of a locally product Riemannian manifold. The equality case is also considered.

Roth (Bull. Aust. Math. Soc. 95, 495-499 (2017)) gave a result for DDVV inequality submanifolds of warped product manifolds. Then Murathan et al. (J. Geom. (2018) 109:30) obtained the Wintgen-like inequality for statistical submanifolds of statistical warped product manifolds. Recently, Gorunus et al. (Inter. Elec. J. Geom. 12(1), 43-56 (2019)) established the generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu statistical manifolds. Thus, in the present manuscript, we find the generalized Wintgen inequality for a Legendrian submanifold in the standard warped product manifolds (β-Kenmotsu manifold).

The aim of the current article is to formulate sufficient conditions for the Laplacian and a gradient of the warping function of a compact warped product submanifold ${\Sigma }^{{\beta }_1+{\beta }_2}$ in a unit sphere ${\mathbb{S}}^d$ that provides trivial homology and fundamental groups. We also validate the instability of current flows in ${\pi }_1\left({\Sigma }^{{\beta }_1+{\beta }_2}\right)$. The constraints are also applied to the warped function eigenvalues and integral Ricci curvatures.

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms ${\mathbb{D}}^{2n+1}\left(ϵ\right)$ and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants ($\delta$-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern.