In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) surfaces up to diffeomorphism.
This makes me wonder about classifying compact Riemannian 2-manifolds up to isometry. In particular:
Is there a classification of all Riemannian 2-manifolds that are diffeomorphic to the 2-sphere?
I imagine this to be a very difficult question. As such, I have two follow-up questions:
If this is a tractable question, how much progress has been made in this direction? What is known and what isn't?
If the question is considered too difficult to have a real answer (is this the case?), then I imagine there to be simpler, related questions. What are some examples of these?