Condition metrics in the three classical spaces

TL;DR

This paper investigates the behavior of geodesics in the condition metric on different classical spaces, proving a positive result for spheres, but providing a counterexample for hyperbolic space.

Contribution

It extends the understanding of geodesic behavior in the condition metric from Euclidean space to spheres and highlights limitations with hyperbolic space.

Findings

Geodesics in the condition metric on spheres have closest points at endpoints.

Counterexample shows the property fails in hyperbolic space.

The property holds under certain smoothness conditions in Euclidean space.

Abstract

Let $(\mathcal{M},g)$ be a Riemannian manifold and $\mathcal{N}$ a $\mathcal{C}^2$ submanifold without boundary. If we multiply the metric $g$ by the inverse of the squared distance to $\mathcal{N}$, we obtain a new metric structure on $\mathcal{M}\setminus\mathcal{N}$ called the condition metric. A question about the behaviour of the geodesics in this new metric arises from the works of Shub and Beltr\'an: is it true that for every geodesic segment in the condition metric its closest point to $\mathcal{N}$ is one of its endpoints? Previous works show that the answer to this question is positive (under some smoothness hypotheses) when $\mathcal{M}$ is the Euclidean space $\mathbb{R}^n$. Here we prove that the answer is also positive for $\mathcal{M}$ being the sphere $\mathbb{S}^n$ and we give a counterexample showing that this property does not hold when $\mathcal{M}$ is the hyperbolic space $\mathbb{H}^n$.