Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

Hülya Aytimur,Kazuhiko Takano,Cihan Özgür,Adela Mihai,Mayuko Kon

doi:10.3390/math7121202

Abstract

We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ ( 2 , 2 ) .

Highlights

In [1], the notion of a statistical manifold was defined by Amari
Mihai studied statistical submanifolds in statistical manifolds of constant curvature and proved inequalities for the scalar curvature and the Ricci curvature associated with the dual connections
The same authors obtained in [10] a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature

Summary

Introduction

In [1], the notion of a statistical manifold was defined by Amari. It has applications in information geometry, which represents one of the main tools for machine learning and evolutionary biology. Mihai studied statistical submanifolds in statistical manifolds of constant curvature and proved inequalities for the scalar curvature and the Ricci curvature associated with the dual connections. The same authors obtained in [10] a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. In their paper, another definition of the sectional curvature, due to Opozda, given in [11], was used (see [12]). Mihai established the Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. In the present article, motivated by the above studies, we obtain a Chen first inequality and an inequality for the Chen δ(2, 2) invariant for statistical submanifolds in Kähler-like statistical manifolds. For our study, we would like to point out that, by referring to the papers [15,16,17], the curvature invariants of statistical submanifolds in Kähler-like statistical manifolds will be investigated

Preliminaries

An Example of a Submanifold of a Kähler-Like Statistical Manifold

A Chen First Inequality