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We study the family of $\alpha$-connections of Amari-Chentsov on thehomogeneous space $\mathcal{D}(M)/\mathcal{D}_\mu(M)$ of diffeomorphisms modulovolume-preserving diffeomorphims of a compact manifold $M$. We show that insome cases their geodesic equations yield completely integrable Hamiltoniansystems.

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