Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

TL;DR

This paper investigates the vanishing of geodesic distances on spaces of submanifolds and diffeomorphism groups, showing that certain metrics induce a meaningful topology despite the $L^2$-metric's distance collapsing to zero.

Contribution

It introduces a new metric involving mean curvature that yields a non-vanishing geodesic distance, addressing the limitations of the $L^2$-metric.

Findings

The $L^2$-metric induces zero geodesic distance on submanifolds and diffeomorphism groups.

A curvature-involving metric provides a positive geodesic distance, serving as a good topological metric.

Vanishing geodesic distance phenomenon also occurs for all diffeomorphism groups under the $L^2$-metric.

Abstract

The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the $L^2$-metric.