Equiaffine Structure and Conjugate Ricci-symmetry of a Statistical Manifold

TL;DR

This paper investigates conditions under which statistical manifolds possess an equiaffine structure, focusing on conjugate Ricci-symmetry as a weaker condition than conjugate symmetry, and explores when these conditions coincide.

Contribution

It introduces conjugate Ricci-symmetry as a new, weaker condition ensuring an equiaffine structure on statistical manifolds and analyzes when it aligns with conjugate symmetry.

Findings

Conjugate Ricci-symmetry implies an equiaffine structure.

Conditions under which conjugate symmetry and conjugate Ricci-symmetry coincide.

Relationship between dual flatness, conjugate symmetry, and equiaffine structures.

Abstract

A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical manifold, which is conjugate Ricci-symmetric, has an equiaffine structure is given, where conjugate Ricci-symmetry is weaker condition than conjugate symmetry. A condition for conjugate symmetry and conjugate Ricci-symmetry to coincide is also given.