Riemannian orthonormal frame ($g(X_i,X_j)={\delta }_{ij}$)

A smooth local basis of vector fields whose inner products under a Riemannian metric are the identity matrix. Tracing the Riemannian Hessian in such a frame gives the Laplace-Beltrami operator, with the chosen sign.

At any point of a Riemannian manifold, choose geodesic normal coordinates and orthonormalize their coordinate fields smoothly. At the point, the metric is the identity and its first derivatives vanish, so the orthonormalization coefficients have zero first derivatives. The Christoffel symbols also vanish there. This gives a local Riemannian orthonormal frame with the displayed property, useful for invariant identities evaluated at one point.