I know the uniqueness of a geodesic connecting two points on a simply connected manifold $M$ with non-positive sectional curvature can be proven easily with Cartan-Hadamard theorem under the assumption that $M$ is complete. How can I go about proving this statement if the completeness of $M$ is NOT assumed?
To clarify, I'm trying to prove when $M$ is a manifold as above (simply connected, nonpositive sectional curvatures, but NOT necessarily complete), then there can be at most one geodesic between any two distinct points.