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

TL;DR

This paper investigates the properties of geodesic distances induced by fractional order Sobolev metrics on the diffeomorphism group, revealing conditions under which the distance vanishes.

Contribution

It extends the understanding of right invariant Sobolev metrics of fractional order on diffeomorphism groups, focusing on geodesic distance behavior.

Findings

Geodesic distance vanishes for 0 ≤ s < 1/2 on compactly supported diffeomorphisms.

Identifies the threshold Sobolev order where geodesic distance becomes non-trivial.

Provides mathematical characterization of geodesic distances in fractional Sobolev metrics.

Abstract

The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold $M$ of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric $H^s$ of order $0\le s<\tfrac12$ on the Lie algebra $\mathfrak X_c(M)$ of vector fields with compact support.