Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

Abstract

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 R × f N 2 and N 1 × f 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.

Geometric Inequalities of Warped Product Submanifolds and Their Applications

“…This paper shows the relation between the notion of warped product manifold and homotopy-homology theory. Therefore, we hope that this paper will be of great interest with respect to the topology of Riemannian geometry [28][29][30][31][32][33][34][35] which may find possible applications in physics.…”

Affine Ricci solitons of three-dimensional Lorentzian Lie groups

“…By the fifth equation in (3.13), we get λ 3 = α + λ. By the sixth equation in (3,13), we get α 2 + λα + λ 2 = 0. Then λ = α = 0, this is a contradiction.…”

Extremities for statistical submanifolds in Кenmotsu statistical manifolds

“…Recently, Siddiqui et al studied statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, they proved Chen's inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi-Civita connection) in [17].…”