TITLE: Cartan-Hadamard's Theorem, Completeness
QUESTION [13 upvotes]: Theorem: Let $(M^n,g)$ complete, simply connected Riemannian manifold with non-positive sectional curvature. Then $M$ is diffeomorphic to $\mathbb{R}^n$.
Question: Can we change complete to "every two points can be joined by a minimizing geodesic"?

REPLY [2 votes]: Let $X$ be completion of $(M,g)$.
Note that $X$ is simply connected.
It follows since any loop in $X$ is a limit of a broken geodesic in $(M,g)$.
Therefore $X$ is CAT(0), in particular any two points are joined by unique geodesic.
Therefore $M$ is diffeomorphic to $\mathbb R^n$ as any star-shaped domain (thanks to alvarezpaiva).