Why is the coordinate basis never locally orthonormal in curved spacetime

Question

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.

Answer 1

The excerpt from Carroll refers to the fact that if we write a general metric field as $g = \sum_{jk}g_{jk}(x)dx^j\otimes dx^k$, then we have $g(\partial_j,\partial_k) = g_{jk}(x)$, so the coordinate vector fields $\partial_k := \partial/\partial x_k$ are orthonormal in the given neighborhood only if $g_{jk}(x)=\delta_{jk}$ in the given neighborhood. In other words, they are orthonormal everywhere in the given neighborhood only if the metric is flat in that neighborhood.

I'm using the language of differential forms, where $dx^k$ is a one-form defined such that $dx^k(\partial_j)=\delta^k_j$.