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

Abstract

We study the geodesic distance induced by right-invariant metrics on the group ${\mathrm{Diff}}_{\text{c}}(M)$ 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 $s\ge 1$, 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. [BBHM13].

arXiv commentVersion 2: some changes in presentation of the paper, no mathematical changes. Now includes a link to some related videos in http://www.math.toronto.edu/rjerrard/geo_dist_diffeo/vanishing.html

References in corpus (5)

• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
• The Lagrangian Radon Transform and the Weil representation
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II
• Vanishing distance phenomena and the geometric approach to SQG

Cited by in corpus (7)

• Vanishing distance phenomena and the geometric approach to SQG
• Fractional Sobolev metrics on spaces of immersions
• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics
• The $L^p$-Fisher-Rao metric and Amari-Cencov $\alpha$-connections
• Generalized compressible flows and solutions of the H(div) geodesic problem
• Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents
• Regularity and completeness of half-Lie groups