Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

Aliya Siddiqui,Oğuzhan Bahadır,Bang-Yen Chen

doi:10.3390/math7090797

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.

Highlights

Statistical manifolds were introduced in 1985 by S
Amari [1] in terms of information geometry, and they were applied by Lauritzen in [2]
Inspired by historical development on the classifications of Casorati curvatures and Ricci curvatures, we establish in Section 6 an inequality for statistical warped product submanifolds in a statistical manifold of constant curvature

Summary

Introduction

Statistical manifolds were introduced in 1985 by S. Sahin [10] studied and obtained the Wintgen-like inequality for statistical submanifolds of statistical warped product manifolds. Our solution to this problem given is derived via the fundamental equations of statistical submanifolds. An extrinsic curvature of a Riemannian submanifold was defined by Casorati in [13], as the normalized square of the length of the second fundamental form. Inspired by historical development on the classifications of Casorati curvatures and Ricci curvatures, we establish in Section 6 an inequality for statistical warped product submanifolds in a statistical manifold of constant curvature.

Basics on Statistical Warped Product Manifolds

Statistical Solitons on Statistical Warped Product Manifolds

Optimal Casorati Inequality

Examples