I am learning the basics of Riemannian Geometry and my teacher invokes multiple times the following proposition :
For any $p \in M$, there exists an open subset $U \ni p$ with the property that any two points $\alpha, \beta$ in $U$ are linked by a unique distance-minimizing geodesic in $M$.
Here, $M$ is a $m$-dimensional connected Riemannian manifold. (Maybe my teacher also assumes $M$ is complete, I am not sure).
My question is : why is this true ? We proved two related things in class : the statement about existence is true if $\alpha = p$ (proved by considering the exponential map $exp_p : \Omega_p \subseteq TM_p \to M$ - but we did not show uniqueness) ; and we saw the Hopf-Rinow theorem. How do we put the pieces together ?