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

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.