A note on exponential varieties, statistical manifolds and Frobenius structures
Abstract
In this paper we consider a class of manifolds corresponding to statistical models, related to exponential families. Exponential manifolds have been considered from the point of view of information geometry and independently real algebraic geometry. However, investigations on this object have evolved independently on both sides, creating a certain gap. The aim of this paper is to unify those both approaches, by first establishing a dictionary between them, and secondly connecting them by introducing our web algebraization theorem, for the case of Frobenius statistical manifolds. This theorem relies on the theory of threewebs, introduced and developed by Blaschke and Cartan. We prove that the 3webs of Frobenius exponential manifolds are algebraizable, i.e. that the $r$dimensional foliations of the web belong to a hypercubic, proving a conjecture of Amari.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.02607
 Bibcode:
 2021arXiv211002607C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry