  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 $\mathbb{H}^n$.
