Riemannian manifold is metric space
where the modulus sign is the norm in the tangent space with respect to the Riemannian metric. The distance between is defined as the infimum of the lengths of all paths from to .
A complete Riemannian manifold, or geodesically complete Riemannian manifold, is one where at any point, the exponential map from the tangent space at the point to the manifold, is well-defined. Complete Riemannian manifolds give rise to complete metric spaces by the Hopf-Rinow theorem. The metric spaces they give rise to are also geodesic metric spaces. Not every geodesic metric space arises from a complete Riemannian manifold (in particular, for instance, the open disc is far from complete but it is a geodesic metric space).
Given a Riemannian manifold and a Riemannian submanifold the metric space structure arising from the Riemannian metric on the submanifold, is different from the metric space structure arising from the Riemannian metric on the manifold, inherited to the submanifold.