Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, and an invariant metric implies that the structure group of the frame bundle is the orthogonal group SO (m, n).
Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group SO (m, n).