  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.
