Curvature Invariants for Statistical Submanifolds of Hessian Manifolds of Constant Hessian Curvature

Adela Mihai,Ion Mihai

doi:10.3390/math6030044

Abstract

We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifold.

Highlights

It is well-known that curvature invariants play the most fundamental role in Riemannian geometry
Curvature invariants provide the intrinsic characteristics of Riemannian manifolds which affect the behavior in general of the Riemannian manifold
In [3], Aydin and the present authors obtained geometrical inequalities for the scalar curvature and the Ricci curvature associated to the dual connections for submanifolds in statistical manifolds of constant curvature

Summary

Introduction

It is well-known that curvature invariants play the most fundamental role in Riemannian geometry. Chen [1] established a generalized Euler inequality for submanifolds in real space forms. A sharp relationship between the Ricci curvature and the squared mean curvature for any Riemannian submanifold of a real space form was proved in [2], which is known as the Chen-Ricci inequality. In [3], Aydin and the present authors obtained geometrical inequalities for the scalar curvature and the Ricci curvature associated to the dual connections for submanifolds in statistical manifolds of constant curvature. It is known [6] that such a manifold is a statistical manifold of null constant curvature and a Riemannian space form of constant sectional curvature −c/4 (with respect to the sectional curvature defined by the Levi-Civita connection)

Statistical Manifolds and Their Submanifolds

Euler Inequality and Chen-Ricci Inequality