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

Robert L. Jerrard; Cy Maor

doi:10.1007/s10455-018-9644-y

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 journal › Article › peer-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 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 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 language	English
Pages (from-to)	631-656
Number of pages	26
Journal	Annals of Global Analysis and Geometry
Volume	55
Issue number	4
DOIs	https://doi.org/10.1007/s10455-018-9644-y
State	Published - 1 Jun 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

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10.1007/s10455-018-9644-y

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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).",

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T1 - Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

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AU - Maor, Cy

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N2 - 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).

AB - 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).

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KW - Fractional Sobolev spaces

KW - Infinite dimensional geometry

KW - Vanishing geodesic distance

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JO - Annals of Global Analysis and Geometry

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

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