The Cartan-Hadamard theorem says that if $M$ is a complete manifold with non-positive sectional curvature, then its universal cover is diffeomorphic to a Euclidean space.
Presumably I've misunderstood something, because this suggests that hyperbolic space has a universal cover that's diffeomorphic to the plane. But surely that can't be the case or else hyperbolic space wouldn't need to be mentioned explicitly in the uniformization theorem.
Where have I gone wrong?
Thanks