TITLE: Lie bracket of local orthonormal basis of vector fields
QUESTION [7 upvotes]: Let $(M, g, \nabla)$ be a Riemannian manifold (with $\nabla$ the Levi-Civita connection of $g$), and let {$E_i$} be a local orthonormal basis of vector fields ($g(E_i, E_j) = \delta_{ij} )$.
What can I say about 
$[E_i, E_j]$ or equivalently (using the torsion-free property of Levi-Civita connection) about $\nabla_{E_i} E_j$? Is it true that $[E_i, E_j]$ vanishes?

REPLY [2 votes]: The commutator doesn't have to vanish, even if the metric is flat. For example, the calculations here show that $[\hat{r}, \hat{\theta}] = \frac{1}{r} \hat{\theta}$ when 
$$ \hat{r} = \cos \theta \hat{x} + \sin \theta \hat{y},\\
\hat{\theta} = -\sin \theta \hat{x} + \cos \theta \hat{y} $$
is the local orthonormal frame associated to the polar coordinate system on $\mathbb{R}^2 \setminus \{ 0 \}$ (with the standard metric).