F-Manifolds and geometry of information

Abstract

The theory of (Formula presented.) -manifolds, and more generally, manifolds endowed with commutative and associative multiplication of their tangent fields, was discovered and formalised in various models of quantum field theory involving algebraic and analytic geometry, at least since the 1990s. The focus of this paper consists in the demonstration that various spaces of probability distributions defined and studied at least since the 1960s also carry natural structures of (Formula presented.) -manifolds. This fact remained somewhat hidden in various domains of the vast territory of models of information storing and transmission that are briefly surveyed here.