Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

By Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. 2011

51 citations recorded by the source corpus.

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diffc(M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S1, the geodesic distance on Diffc(S1) vanishes if and only if $${s\leq\frac12}$$. For other manifolds, we obtain a partial characterization: the geodesic distance on Diffc(M) vanishes for $${M=\mathbb{R}\times N, s < \frac12}$$ and for $${M=S^1\times N, s\leq\frac12}$$, with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diffc(M) is positive for $${{\rm dim}(M)=1, s > \frac12}$$ and dim(M) ≥ 2, s ≥ 1. For $${M=\mathbb{R}^n}$$, we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for $${s=\frac12}$$, and the Camassa–Holm equation for s = 1.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fc57d7da207a3a07ace12240072a071de4e55d9b