Relevant Differential Topology in Statistical Manifolds

Abstract

The author agrees and assumes that this paper is a mixture of several subjects and ongoing perspectives but he emphasizes that their unifying framework is formed of the trio described in Prologue. Statistical models form a subcategory of the category of statistical manifolds. Intuitively, think of the notion of Foliation as (geometrical or topological) study of (discrete, continuous or differentiable) partitions. The Geometry of Statistical manifolds provides a unified framework for these studies. From the conceptual point of view the statistical geometry highlights many bridges betwen the Riemannian Geometry and the Cartan geometry or gauge geometry, (which may be understood nowadays as the studies of Koszul connections). This paper is devoted to point out some relevant invariants of the differential topology which are linked with the structure of the statistical manifold. We aim to point out that a structure of statistical manifold yields many interesting dynamical systems and many relevant structured foliations, i.e. foliations whose leaves unformally carry a prescribed geometric structure.