Cartan-Hadamard's Theorem, Completeness 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"?
 A: Take a point $x \in M$ and consider the star-shaped open set  $U \subset T_xM$ where the exponential
$$
exp_x : U \rightarrow M
$$
is defined. $U$ is diffeomorphic to ${\mathbb R}^n$ and the map ${\rm exp_x}$ is
(1) a local diffeomorphism (because of the absence of conjugate points = singularities of ${\rm exp}_x$);
(2) surjective (because I can join any other point of $M$ to $x$ by a geodesic).
I guess there is no problem and $M$ is diffeomorphic to $U$. Right??
Edit. As Claudio points out, the fact that ${\rm exp}_x$ is surjective and a local diffeomorphism does not imply that it is a covering map. Hence we cannot use the homotopy lifting property to conclude that it is injective and a diffeomorphism.
A: 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).
