1. The problem statement, all variables and given/known data Suppose that for every smooth Riemannian metric on a manifold M, M is complete. Show that M is compact. 2. The attempt at a solution I'm honestly not too sure how to start this question. If we could show that the manifold is totally bounded we would be done, but I'm not sure how to get that out of the assumption. Any ideas that I could play around with?