On the manifoldM(M) of all Riemannian metrics on a compact manifold M, one can consider the natural L2-metric as described
first by [11]. 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.