Self-convexity and curvature

Let (M,g)(M,g) be a Riemannian manifold and NN a C2C2 submanifold without boundary. If we multiply the metric gg by the inverse of the squared distance to NN, we obtain a new metric structure on M∖NM∖N called the condition metric  . A question about the behavior of this new metric arises from the works of Beltrán, Dedieu, Malajovich and Shub: is it true that for every geodesic segment in the condition metric its closest point to NN is one of its endpoints? Previous works show that the answer to this question is positive (under some smoothness hypotheses) when MM is the Euclidean space RnRn. Here we find that there is a strong relation between this property and the curvature of MM. In particular, we prove that the answer is also positive if MM has non-negative curvature and that this property does not hold when the curvature of MM is strictly negative at some point.