On Completeness of Groups of Diffeomorphisms

Abstract

We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or compact without boundary. The main result is that for $s > \dim M/2 + 1$, the group $\mathcal D^s(M)$ is geodesically and metrically complete with a surjective exponential map. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.