A manifold possessing a metric tensor. For a complete Riemannian manifold, the metric d(x, y) is defined as the length of the shortest curve (geodesic) between x and y. Every complete Riemannian manifold is boundedly compact. This is part of or a consequence of the Hopf-Rinow theorem.

Bishop's inequality | Campbell's theorem | Cheeger's finiteness theorem | Hopf-Rinow theorem | pseudo-Riemannian manifold | Riemannian metric