A smooth manifold \(M\) equipped with Riemannian metric \(g\) is called Riemannian manifold. We usually denote it by \((M,g)\).
Remark
- Every [smooth manifold](smooth manifold.md) admits a Riemannian metric and it is therefore a Riemannian manifold [@lee2013smooth_manifolds, Proposition 13.3]. But the choice is not unique.