  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.
