## Abstract

We study the geodesic distance induced by right-invariant metrics on the group Diff _{c}(M) of compactly supported diffeomorphisms of a manifold M and show that it vanishes for the critical Sobolev norms W^{s} ^{,} ^{n} ^{/} ^{s}, where n is the dimension of M and s∈ (0 , 1). This completes the proof that the geodesic distance induced by W^{s} ^{,} ^{p} vanishes if sp≤ n and s< 1 , and is positive otherwise. The proof is achieved by combining the techniques of two recent papers—(Jerrard and Maor in Ann Glob Anal Geom 55(4):631–656, 2019) by the authors, which treated the subcritical case, and Bauer et al. (Vanishing distance phenomena and the geometric approach to SQG, 2018. arXiv:1805.04401), which treated the critical one-dimensional case.

Original language | English |
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Pages (from-to) | 351-360 |

Number of pages | 10 |

Journal | Annals of Global Analysis and Geometry |

Volume | 56 |

Issue number | 2 |

DOIs | |

State | Published - 1 Sep 2019 |

Externally published | Yes |

### Bibliographical note

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

## Keywords

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