A remark on vanishing geodesic distances in infinite dimensions

TL;DR

This paper discusses how a weak Riemannian metric in a Hilbert manifold can lead to a vanishing geodesic distance, highlighting a peculiar geometric phenomenon in infinite-dimensional spaces.

Contribution

It demonstrates the construction of a vanishing geodesic distance from a weak Riemannian metric in infinite-dimensional Hilbert manifolds.

Findings

Vanishing geodesic distance can occur in infinite-dimensional Hilbert manifolds.

Weak Riemannian metrics can induce degenerate geodesic distances.

The paper provides a method to construct such vanishing distances.

Abstract

We observe that a vanishing geodesic distance arising from a weak Riemannian metric in a Hilbert manifold can be constructed.