Sobolev metrics on the manifold of all Riemannian metrics

On the manifold $\Met(M)$ of all Riemannian metrics on a compact manifold $M$ one can consider the natural $L^2$-metric as described first by \cite{Ebin70}. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.