Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $Amapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Delta^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.