Riemannian geometry on spaces of submanifolds induced by the diffeomorphism group

The space of embedded submanifolds plays an important role in applications such as computational anatomy and shape analysis. We can define two different classes on Riemannian metrics on this space: so-called outer metrics are metrics that measure shape changes using deformations of the ambient space and they find applications mostly in computational anatomy; the second class that are defined directly on the space of embeddings using intrinsic differential operations and they are used in shape analysis. In this paper we compare for the first time the topologies and the geodesic distance functions induced by these the two classes of metrics.