  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.
