• DOI:10.1007/s10455-012-9353-x
• Corpus ID: 6728215

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

@article{Bauer2011GeodesicDF,
  title={Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group},
  author={Martin Bauer and Martins Bruveris and Philipp Harms and Peter W. Michor},
  journal={Annals of Global Analysis and Geometry},
  year={2011},
  volume={44},
  pages={5-21}
}

• Martin Bauer, M. Bruveris, +1 author P. Michor
• Published 2 May 2011
• Mathematics
• Annals of Global Analysis and Geometry

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.