There seems to be a lot of theorems allowing to prove restricted cases of this (eg. uniformization, classification theorem for compact surfaces) . Intuitively, it seems true, but I've never seen a proof of the general case.



Do all surfaces (2d riemanian manifolds) admit constant curvature?
This is a standard consequence of the uniformization, but I agree that locating a reference may be a challenge. You can find details in my survey <http://arxiv.org/abs/1306.1256>, see Theorem 2.2 where it is shown that any open (smooth connected) 2-manifold admits a complete metric of constant negative curvature. The same proof works for compact manifolds but in this case you need to allow for two other possibilities of constant curvature 0 and 1, and it is clear from the proof why this is the case.


