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!

No, none. You are describing space forms, in this case the hyperbolic plane and 3-space. You might want to look at Cheeger and Ebin, Comparison Theorems in Riemannian Geometry.

Right, Theorem 1.37 on page 41, simply connected manifolds of the same dimension and constant (sectional) curvature $K$ are isometric. Corollary of Cartan-Ambrose-Hicks.

Meanwhile, you get the compact surfaces precisely because there are Fuchsian groups, acting on $\mathbb H^2,$ and your surfaces are quotient spaces. It turns out that Fuchsian groups, and the flower and color Fuchsia, are named after different people named Fuchs. Go figure.