In Carrolls GR book “Spacetime and Geometry” he comments that
“This is not a situation we can define away; on a curved manifold, a co- ordinate basis will never be orthonormal throughout a neighborhood of any point where the curvature does not vanish.”
Is there a clear proof that the coordinate basis cannot be orthonormal in a curved geometry except for potentially at one point in space?
My intuition says it is because curvature can either make the metric non-diagonalizable negating orthogonality, or it stretches basis vectors negating normality but its not clear this is correct.