Condition metrics in the three classical spaces

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 Beltrán: 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 $\mathbb{H}^n$.

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