Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature.

Simona Decu,Stefan Haesen,Leopold Verstraelen,Gabriel-Eduard Vîlcu

doi:10.3390/e20070529

Abstract

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant -sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and of the submanifold (associated with the dual connections) satisfy , i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection.

Highlights

A fundamental problem in the general theory of Riemannian submanifolds is to establish simple relationships between the main intrinsic invariants and the main extrinsic invariants of the submanifolds [1]. Such simple relationships can be provided by certain types of inequalities
The Chen ideal submanifolds have been investigated, i.e., the submanifolds that do realize an optimal equality in Chen inequalities
Lee et al established optimal inequalities involving the Casorati curvatures and the normalized scalar curvature on submanifolds of statistical manifolds of constant curvature [42]. These inequalities were extended by Aquib and Shahid [43] in the setting of statistical submanifolds in quaternion Kähler-like statistical space forms

Summary

Introduction

A fundamental problem in the general theory of Riemannian submanifolds is to establish simple relationships between the main intrinsic invariants and the main extrinsic invariants of the submanifolds [1]. Lee et al [20] studied optimal inequalities in terms of δ-Casorati curvatures of submanifolds in Kenmotsu space forms. Lee et al established optimal inequalities involving the Casorati curvatures and the normalized scalar curvature on submanifolds of statistical manifolds of constant curvature [42]. These inequalities were extended by Aquib and Shahid [43] in the setting of statistical submanifolds in quaternion Kähler-like statistical space forms. We establish inequalities in terms of the extrinsic normalized δ-Casorati curvatures and the intrinsic scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant φ-sectional curvature. If the bilinear form A defined by Equation (14) is positive semi-definite on the submanifold M, the critical points of f | M, which coincide with the points where the gradient of f is normal to M, are global optimal solutions of the problem (13) [48]

Results’ Main Inequalities

An Example and Concluding Remarks

Conclusions