Given a Riemannian manifold \((M,g)\). A local frame \((E_1, \ldots, E_n)\) is called orthonormal if at each point \((E_1,\ldots,E_n)\) is an orthonormal system regarding \(g\).

Remark
  • On a smooth manifold for every point exists a smooth orthonormal frame in the neighbourhood [@lee2013smooth_manifolds, Corollary 13.8].