Is the hyperbolic plane the only simply connected hyperbolic 2-manifold?

Question

Let $S$ be a simply connected Riemannian 2-manifold with everywhere negative curvature. Is $S$ necessarily diffeomorphic to $\mathbb{R}^2$?

Answer 1

Yes, the geometric classification of surfaces tells us that a simply connected Riemannian surface $S$ must be (up to diffeomorphism) the sphere $S^2$, the complex plane $\mathbf{C}$, or the hyperbolic plane $\mathbf{H}$. Given that $\mathbf{H}$ is the only one of these with negative curvature, $S$ must be the hyperbolic plane.

Answer 2

Yes, by Uniformization theorem