Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

  • Robert L. Jerrard
  • Cy MaorEmail author


We study the geodesic distance induced by right-invariant metrics on the group \({\text {Diff}}_\text {c}({\mathcal {M}})\) of compactly supported diffeomorphisms, for various Sobolev norms \(W^{s,p}\). Our main result is that the geodesic distance vanishes identically on every connected component whenever \(s<\min \{n/p,1\}\), where n is the dimension of \({\mathcal {M}}\). We also show that previous results imply that whenever \(s > n/p\) or \(s \ge 1\), the geodesic distance is always positive. In particular, when \(n\ge 2\), the geodesic distance vanishes if and only if \(s<1\) in the Riemannian case \(p=2\), contrary to a conjecture made in Bauer et al. (Ann Glob Anal Geom 44(1):5–21, 2013).


Diffeomorphism group Vanishing geodesic distance Fractional Sobolev spaces Infinite dimensional geometry 



We are grateful to Meital Kuchar for her help with the figures and to the anonymous referee for their helpful comments. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada under operating Grant 261955.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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