Locally conformally Hessian and statistical manifolds
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;r; g) such that the monodromy group acts on
C by Hessian homotheties, i.e. this action preserves r 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 is
Killing and satisfies r = 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.