Negative curvature compact manifolds I know there is a theorem about the existence of metrics with constant negative curvature  in compact orientable surfaces with genus greater than 1. My intuition of the meaning of genus make me think that surfaces with genus greater that 1 cannot be simply-connected, but as my knowledge about algebraic-topology is zero, I might be wrong.
My question is: are there two and three dimensional orientable compact manifolds with constat negative curvature that are simply-connected? If yes, what is an example of one? If not, what is the reason?
Thanks in advance!

Every two (classification of surfaces) and three (Poincare-Thurston-Perelman) closed simply-connected manifold is diffeomorphic to a sphere, hence does not admit any metric of negative curvature.
If a manifold has constant sectional curvature, its metric lifts to the universal cover, which is isometric to one of the space forms. Since a simply connected manifold is homeomorphic to its universal cover, and hyperbolic space is non-compact, no closed negatively curved manifold is simply connected.
