Two dimensional spacetime and the Gauss Bonnet theorem

Question

Generally two dimensional spacetimes are deemed to be static, as the Gauss Bonnet theorem implies that the Einstein Hilbert action would be a constant independent of $g$.

But as far as I can tell, the Gauss Bonnet theorem only applies to compact manifolds, even in the version for Lorentzian manifolds. The extension to non compact manifold is only an inequality, which does not specify that the result has to be independent of the metric.

Are two dimensional manifolds actually static as far as the metric goes, and if so, how to prove it?

This post imported from StackExchange Physics at 2015-11-21 21:51 (UTC), posted by SE-user Slereah

Comments

Should it not suffice, that the action is constant on all compact subsets of the manifold? Then you get the result for each compact subset, which say that subset is static. As you can cover your manifold with compact sets this implies the result.

This post imported from StackExchange Physics at 2015-11-21 21:51 (UTC), posted by SE-user Sebastian Riese