Riemannian geometry on spaces of submanifolds induced by the diffeomorphism group

TL;DR

This paper compares two types of Riemannian metrics on the space of embedded submanifolds, analyzing their topologies and geodesic distances to advance understanding in shape analysis and computational anatomy.

Contribution

It provides the first comparison of topologies and geodesic distances induced by outer and intrinsic Riemannian metrics on submanifold spaces.

Findings

Outer and intrinsic metrics induce different topologies.

Geodesic distances vary significantly between the two metric classes.

Results inform applications in shape analysis and computational anatomy.

Abstract

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.