Locally conformally Hessian and statistical manifolds

Abstract

A statistical manifold (M,D,g) is a manifold M endowed with a torsion-free connection D and a Riemannian metric g such that the tensor Dg is totally symmetric. If D is flat then (M,g,D) is a Hessian manifold. A locally conformally Hessian (l.c.H.) manifold is a quotient of a Hessian manifold (C,∇,g) such that the monodromy group acts on C by Hessian homotheties, i.e. this action preserves ∇ 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 ξ is Killing and satisfies ∇ξ=λId. 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.