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

Abstract

We study Sobolev-type metrics of fractional order $s\ge 0$ 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 $s\le \frac{1}{2}$. For other manifolds we obtain a partial characterization: the geodesic distance on $\Diff_c(M)$ vanishes for $M=\mathbb{R}\times N,s<\frac{1}{2}$ and for $M=S^1\times N,s\le \frac{1}{2}$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on $\Diff_c(M)$ is positive for $\dim (M)=1,s>\frac{1}{2}$ and $\dim (M)\ge 2,s\ge 1$. For $M={\mathbb{R}}^n$ 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 $s=\frac{1}{2}$ and the Camassa-Holm equation for $s=1$.

16 pages. Final version

References in corpus (3)

• Sobolev metrics on shape space of surfaces
• Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation
• The geometry of a vorticity model equation

Cited by in corpus (17)

• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
• Why Use Sobolev Metrics on the Space of Curves
• A zoo of diffeomorphism groups on ${\mathbb{R}}^n$
• Geometric investigations of a vorticity model equation
• Local and Global Well-posedness of the fractional order EPDiff equation on ${\mathbb{R}}^d$
• The Lagrangian Radon Transform and the Weil representation
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II
• The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line
• Geodesic Completeness for Sobolev $H^s$-metrics on the Diffeomorphisms Group of the Circle
• Vanishing distance phenomena and the geometric approach to SQG
• Fractional Sobolev metrics on spaces of immersions
• Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups
• Sub-Riemannian structures on groups of diffeomorphisms
• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics
• Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents
• Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves