Open Access
June 2013 Sobolev Metrics on the Manifold of All Riemannian Metrics
Martin Bauer, Philipp Harms, Peter W. Michor
J. Differential Geom. 94(2): 187-208 (June 2013). DOI: 10.4310/jdg/1367438647


On the manifold $\mathcal{M}(M)$ of all Riemannian metrics on a compact manifold $M$, one can consider the natural $L^2$-metric as described first by D.G. Ebin, The manifold of Riemannian metrics. 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.


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Martin Bauer. Philipp Harms. Peter W. Michor. "Sobolev Metrics on the Manifold of All Riemannian Metrics." J. Differential Geom. 94 (2) 187 - 208, June 2013.


Published: June 2013
First available in Project Euclid: 1 May 2013

zbMATH: 1275.58007
MathSciNet: MR3080480
Digital Object Identifier: 10.4310/jdg/1367438647

Rights: Copyright © 2013 Lehigh University

Vol.94 • No. 2 • June 2013
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