The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point.
It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898.
Élie Cartan generalized the theorem to Riemannian manifolds in 1928 (Helgason 1978; do Carmo 1992; Kobayashi & Nomizu 1969).
The convexity of the distance function along a pair of geodesics is a well-known consequence of non-positive curvature of a metric space, but it is not equivalent (Ballmann 1990).
The metric form of the theorem demonstrates that a non-positively curved polyhedral cell complex is aspherical.