A Riemannian manifold is known to be complete if and only if any of its closed bounded subset is compact.
Is there a similar criterion for Riemannian manifolds for which every bounded subset is totally bounded with respect to the geodesic distance, that is, every bounded set for the geodesic distance can be covered by finitely many geodesic balls of arbitrarily small radius?
Obviously this will happen in any complete Riemannian manifold.
There are manifolds that are bounded without being totally bounded, see https://math.stackexchange.com/q/749891/105127.