Bochner's technique for statistical structures

TL;DR

This paper extends Bochner's technique to statistical structures, introducing new curvature concepts and proving theorems that relate topology, geometry, and statistical connections, thereby advancing the understanding of statistical geometric frameworks.

Contribution

It develops Bochner's technique for statistical structures, introduces a new sectional curvature concept, and proves related global and local geometric theorems.

Findings

Established Bochner-Weitzenbock and Simon's formulas for statistical structures

Proved conditions under which statistical structures are trivial

Introduced a new sectional curvature concept for statistical connections

Abstract

The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorems on the geometry of statistical structures are proved, for instance, theorems saying that under some topological and geometrical conditions a statistical structure must be trivial. We also introduce a new concept of sectional curvature depending on statistical connections. On the base of this notion we study the curvature operator and prove some analogues of well-known theorems from Riemannian geometry.