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 $0\le s<\tfrac12$ on the Lie algebra $\mathfrak X_c(M)$ of vector fields with compact support. 
Martin Bauer, Martins Bruveris, Peter W. Michor
math.DG math.AP