How to show (or refute) that point pair connecting geodesic segments are locally uniqe on a Riemann manifold?

Question

I believe it is correct to say that for every point on a Riemann manifold there is a sufficiently small nice neighborhood within which every point pair determines exactly one connecting geodesic segment.

Is this a correct statement? Is it an axiom or a theorem? If it is a theorem, how is it proved?