Locally conformally Hessian and statistical manifolds

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.