Locally conformally Hessian and statistical manifolds

TL;DR

This paper studies locally conformally Hessian (l.c.H) statistical manifolds, especially radiant ones of rank 1, proving density results and a structure theorem for compact cases involving fibered manifolds with statistical structures.

Contribution

It introduces the concept of radiant l.c.H. manifolds of rank 1, proves their density among all radiant l.c.H. metrics, and provides a detailed structure theorem for compact cases.

Findings

Radiant l.c.H. metrics of rank 1 are dense among all radiant l.c.H. metrics.

Compact radiant l.c.H. manifolds of rank 1 are fibered over a circle with statistical manifold fibers.

The structure of such manifolds is determined by the statistical structure on fibers and monodromy automorphisms.

Abstract

A statistical manifold $\left(M,D,g\right)$ is a manifold $M$ endowed with a torsion-free connection $D$ and a Riemannian metric $g$ such that the tensor $D g$ is totally symmetric. If $D$ is flat then $\left(M,g,D\right)$ is a Hessian manifold. A locally conformally Hessian (l.c.H) manifold is a quotient of a Hessian manifold $(C,\nabla,g)$ such that the monodromy group acts on $C$ by Hessian homotheties, i.e. this action preserves $\nabla$ and multiplies $g$ by a group character. The l.c.H. rank is the rank of the image of this character considered as a function from the monodromy group to real numbers. A l.c.H. manifold is called radiant if the Lee vector field $\xi$ is Killing and satisfies $\nabla \xi =\lambda \Id$. We prove that the set of radiant l.c.H. metrics of l.c.H. rank 1 is dense in the set of all radiant l.c.H. metrics. We prove a structure theorem for compact radiant l.c.H. manifold of l.c.H. rank 1. Every such manifold $C$ is fibered over a circle, the fibers are statistical manifolds of constant curvature, the fibration is locally trivial, and $C$ is reconstructed from the statistical structure on the fibers and the monodromy automorphism induced by this fibration.