I was reading this question and wondering if there are more general results answering the question "Under which condition do manifolds have to have a finite riemannian volume?". I know that bounded balls can have infinite volume even in a riemannian setting.
I would appreciate any hints and result!
On
The Bishop-Gromov inequality states that any finite-radius metric ball on which the Ricci curvature is bounded below, even by a negative constant, has finite volume. So a sufficient condition would be bounded + Ricci curvature bounded below. This still isn't a sufficient condition; there's unbounded manifolds with finite volume.
Myers' theorem gives a sufficient condition: any complete Riemannian manifold whose Ricci curvature is bounded below by a positive constant is compact, and thus has finite volume.