Metric embedding of negatively curved surfaces Suppose a metric surface, simply connected and locally isometric to the hyperbolic plane; Do you can embed this surface on the hyperbolic plane?
The fact is known to be false in the spherical case (that is, there exist a surface, simply connected and locally isometric to the Riemann sphere that does not embed on the Riemann sphere).
I'm not sure that the fact is really true, but I've heard this before, and I need to understand why it's true (if it is at all) and why the proof doesn't work on the spherical case.
Thanks in advance!

For a counterexample, take any point $p \in \mathbb{H}^2$, and take the universal cover of $\mathbb{H}^2 - p$.
A spherical counterexample is similar, except removing a single point does not work because the result is already simply connected and so nothing changes when you take its universal cover. So you simply remove two points $p,q \in S^2$. That is, take the universal cover of $S^2 - \{p,q\}$.
Notice, in these counter-examples the metrics are not complete. If instead you take a complete simply connected surface locally isometric to the hyperbolic plane, then it is globally isometric to the hyperbolic plane. Similarly, a complete simply connected surface locally isometric to $S^2$ is globally isometric to $S^2$.
