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}
}
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… 
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