## Journal of Differential Geometry

### Sobolev Metrics on the Manifold of All Riemannian Metrics

#### Abstract

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.

#### Article information

Source
J. Differential Geom., Volume 94, Number 2 (2013), 187-208.

Dates
First available in Project Euclid: 1 May 2013

https://projecteuclid.org/euclid.jdg/1367438647

Digital Object Identifier
doi:10.4310/jdg/1367438647

Mathematical Reviews number (MathSciNet)
MR3080480

Zentralblatt MATH identifier
1275.58007

#### Citation

Bauer, Martin; Harms, Philipp; Michor, Peter W. Sobolev Metrics on the Manifold of All Riemannian Metrics. J. Differential Geom. 94 (2013), no. 2, 187--208. doi:10.4310/jdg/1367438647. https://projecteuclid.org/euclid.jdg/1367438647