Markov invariant geometry on state manifolds

This paper is devoted to certain differential-geometric constructions in classical and noncommutative statistics, invariant with respect to the category of Markov maps, which have recently been developed by Soviet, Japanese, and Danish researchers. Among the topics considered are invariant metrics and invariant characteristics of informational proximity, and lower bounds are found for the uniform topologies that they generate on sets of states. A description is given of all invariant Riemannian metrics on manifolds of sectorial states. The equations of the geodesies for the entire family of invariant linear connections $\Delta={}^\gamma\Delta$, $\gamma\in\mathbb R$, are integrated on sets of classical probability distributions. A description is given of the protective structure of all the geodesic curves and totally geodesic submanifolds, which turns out to be a local lattice structure; it is shown to coincide, up to a factor $\gamma(\gamma-1)$, with the Riemann–Christoffel curvature tensor.