Sobolev metrics on the Riemannian manifold of all Riemannian metrics

Details Sobolev metrics on the Riemannian manifold of all Riemannian metrics Sobolev metrics on the Riemannian manifold of all Riemannian metrics

2011-02-16

arxiv.org/abs/1102.3347v1

Mathematics

Differential Geometry

14 pages

Scientific paper

On the manifold $\Met(M)$ of all Riemannian metrics on a compact manifold $M$ one can consider the natural $L^2$-metric as decribed first by Ebin, 1970. 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 this metrics.

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