A Riemannian manifold is a smooth manifold equipped with a Riemannian metric.
See also pseudo-Riemannian manifold
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.
For infinite-dimensional manifolds see also orthogonal structure.
The analog in complex geometry is the notion of Kähler manifold.