For a compact manifold $M^m$ we shall consider the space
$Met_{H^\alpha}(M)$ of all Riemannian metrics of Sobolev class $H^\alpha$ for real $\alpha > m/2$. The $L^2$-me$tric on $Met_{C^\infty}(M)$ was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on $Met_{C^\infty}(M)$ were considered in [M.Bauer, P.Harms, and P.W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. J. Differential Geom.,94(2):187-208, 2013.]
The main result in this talk will be that the mapping $g\mapsto(1+\Delta^g)^{(r-s)/2}$ is a smooth mapping $Met_{H^\alpha}(M) \to L(\Gamma_{H^s}(E), \Gamma_{H^r}(E))$ for $r,s\in [-\alpha,\alpha]$, and is real analytic into the larger space $ L(\Gamma_{H^s}(E), \Gamma_{H^{r-\epsilon}}(E))$ for any $\epsilon >0$ so that also $r-\epsilon \in [-\alpha,\alpha]$.
This leads to wellposed-ness of the geodesic equation of the Sobolev $H^\alpha$-metric
$$ G^\alpha_g(h,k) = \int_M Tr(g^{-1}((1+\Delta^g)^{\alpha/2}h)g^{-1}k)vol(g)$$
on $Met_{H^s}(M)$ for each $s\ge \alpha$.
This result is from the recent paper [ Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor: Smoothness of the fractional Laplacian on the space of Riemannian metrics of Sobolev order]
