Annals of Global Analysis and Geometry

, Volume 44, Issue 1, pp 5–21 | Cite as

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

  • Martin Bauer
  • Martins Bruveris
  • Philipp Harms
  • Peter W. Michor
Article

Abstract

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S 1, the geodesic distance on Diff c (S 1) vanishes if and only if \({s\leq\frac12}\). For other manifolds, we obtain a partial characterization: the geodesic distance on Diff c (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 Diff c (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.

Keywords

Diffeomorphism group Geodesic distance Sobolev metrics of non-integral order 

Mathematics Subject Classification (1991)

Primary 35Q31 58B20 58D05 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Martin Bauer
    • 1
  • Martins Bruveris
    • 2
  • Philipp Harms
    • 3
  • Peter W. Michor
    • 1
  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.Insitut de mathématiquesEPFLLausanneSwitzerland
  3. 3.Edlabs, Harvard UniversityCambridgeUSA

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