Riemannian manifold

It is a smooth manifold $(\mathcal{M},\mathcal{O},{\mathcal{A}}_{◻})$ together with a Riemannian metric.
They are, to a certain extent, a kind of metric space. The Hopf-Rinow theorem tells when they are metric spaces (indeed, Cauchy complete metric spaces).

They can be generalized to pseudo-Riemannian manifolds. See the latter entry if you miss something here.