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

Robert L. Jerrard, Cy Maor*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

We study the geodesic distance induced by right-invariant metrics on the group Diff c(M) of compactly supported diffeomorphisms, for various Sobolev norms Ws,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 M. We also show that previous results imply that whenever s> n/ p or s≥ 1 , the geodesic distance is always positive. In particular, when n≥ 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).

Original languageAmerican English
Pages (from-to)631-656
Number of pages26
JournalAnnals of Global Analysis and Geometry
Volume55
Issue number4
DOIs
StatePublished - 1 Jun 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature B.V.

Keywords

  • Diffeomorphism group
  • Fractional Sobolev spaces
  • Infinite dimensional geometry
  • Vanishing geodesic distance

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