A Riemannian manifold is a manifold equipped with a Riemannian metric.
To be distinguished from Riemann surface. In particular, the concepts of 2-dimensional Riemannian manifolds and Riemann surfaces are, while closely related, crucially different.
There is a refinement of topological cobordism categories to one of Riemannian cobordisms.
See also the Myers–Steenrod theorem.