Some inequalities and applications of Simons' type formulas in Riemannian, affine and statistical geometry

TL;DR

This paper develops new formulas and theorems related to curvatures and metric properties in statistical structures, generalizing known results from Lagrangian submanifolds and affine hypersurfaces.

Contribution

It introduces generalized formulas and theorems for statistical structures, extending classical results in differential geometry.

Findings

New inequalities for statistical structures

Generalizations of Simons' formulas in various geometries

Applications to curvature and metric properties

Abstract

A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the case of Lagrangian submanifolds or affine hypersurfaces.