Amari-Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups

TL;DR

This paper investigates Amari-Chentsov $oldsymbol{ extit{ extalpha}}$-connections on homogeneous spaces of diffeomorphism groups, revealing cases where their geodesic equations form completely integrable Hamiltonian systems.

Contribution

It characterizes the geodesic equations of Amari-Chentsov connections on diffeomorphism homogeneous spaces and identifies conditions for their complete integrability.

Findings

Certain geodesic equations are completely integrable Hamiltonian systems

The study extends understanding of geometric structures on diffeomorphism groups

Connections relate to volume-preserving and general diffeomorphisms

Abstract

We study the family of $\alpha$-connections of Amari-Chentsov on the homogeneous space $\mathcal{D}(M)/\mathcal{D}_\mu(M)$ of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold $M$. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.