Cartan-Hadamard states that a non-positively curved Riemannian manifold is covered by $\mathbb{R}^n$. How does it follow that in each homotopy class of paths from $x$ to $y$ there exists a unique geodesic?
I already know that geodesics minimise lengths locally and that the geodesics given by exp$(tv)$ are globally length minimising. Does it follow that any geodesic is of this form?