Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

TL;DR

This paper investigates fractional Sobolev metrics on diffeomorphism groups, characterizing when the induced geodesic distance vanishes or is positive, and relates these metrics to well-known PDEs in hydrodynamics.

Contribution

It provides a partial characterization of geodesic distance behavior for fractional Sobolev metrics on diffeomorphism groups, including explicit PDE connections.

Findings

Geodesic distance vanishes for s ≤ 1/2 on S^1

Geodesic distance is positive for s > 1/2 in 1D

Connections to PDEs like Burgers', Constantin-Lax-Majda, and Camassa-Holm

Abstract

We study Sobolev-type metrics of fractional order $s\geq0$ on the group $\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on $\Diff_c(S^1)$ vanishes if and only if $s\leq\frac12$. For other manifolds we obtain a partial characterization: the geodesic distance on $\Diff_c(M)$ vanishes for $M=\R\times N, s<\frac12$ and for $M=S^1\times N, s\leq\frac12$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on $\Diff_c(M)$ is positive for $\dim(M)=1, s>\frac12$ and $\dim(M)\geq2, s\geq1$. For $M=\R^n$ we discuss the geodesic equations for these metrics. For $n=1$ we obtain some well known PDEs of hydrodynamics: Burgers' equation for $s=0$, the modified Constantin-Lax-Majda equation for $s=\frac 12$ and the Camassa-Holm equation for $s=1$.