Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold

TL;DR

This paper characterizes the Fisher-Rao metric on the space of smooth densities on a compact manifold, deriving its geodesics, curvature, and completeness properties, thus deepening understanding of its geometric structure.

Contribution

It explicitly determines geodesics and curvature for the Fisher-Rao metric, and analyzes completeness, providing new insights into its geometric properties.

Findings

Explicit formulas for geodesics of the Fisher-Rao metric

Curvature properties of the Fisher-Rao metric

Results on geodesic and metric completeness

Abstract

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $$ G_\mu(\alpha,\beta)=C_1(\mu(M)) \int_M \frac{\alpha}{\mu}\frac{\beta}{\mu}\,\mu + C_2(\mu(M)) \int_M\alpha \cdot \int_M\beta $$ for some smooth functions $C_1,C_2$ of the total volume $\mu(M)$. Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness.