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

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 $0\le s<\frac{1}{2}$ on the Lie algebra ${\mathfrak{X}}_c(M)$ of vector fields with compact support.