From Dual Connections to Almost Contact Structures

TL;DR

This paper explores conditions under which a statistical manifold, characterized by a dualistic structure, admits various almost contact and related geometric structures, especially in three dimensions.

Contribution

It establishes criteria linking dualistic structures to the existence of almost contact and related structures on manifolds, expanding understanding in information geometry.

Findings

Conditions for almost contact structures on statistical manifolds

Characterization of contact and cosymplectic structures in 3D

Criteria for coKähler structures in the context of dual connections

Abstract

A dualistic structure on a smooth Riemaniann manifold $M$ is a triple $(M,g,\nabla)$ with $g$ a Riemaniann metric and $\nabla$ an affine connection, generally assumed to be torsionless. From $g$ and $\nabla$, the dual connection $\nabla^*$ can be defined and the triple $(M, \nabla,\nabla^*)$ is called a statistical manifold, a basic object in information geometry. In this work, we give conditions based on this notion for a manifold to admit an almost contact structure and some related structures: almost contact metric,contact, contact metric, cosymplectic, and coK\"ahler in the three-dimensional case.