Negative curvature compact manifolds

Question

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!

Answer 1

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.

Answer 2

Your intuition is correct. In two dimensions, the only simply connected compact orientable surface is the sphere, which must have positive curvature somewhere by Gauss-Bonnet.

In more generality your question is answered by the Cartan-Hadamard theorem: the universal cover of a negatively curved manifold is non-compact, and thus in order for the space itself to be compact it must have non-trivial fundamental group.

Answer 3

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.