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I was doing some calculations with tensors and ran into a result which seems a bit odd to me. I hope someone can validate this or tell me where my mistake is.

So I have a normal orthonormal frame field [itex]\{E_i\}[/itex] in the neighbourhood of a point [itex]p[/itex] in a Riemannian manifold [itex](M,g)[/itex], i.e. Riemannian normal coordinates about [itex]p[/itex] such that [itex]\nabla_{E_i}E_k=0[/itex] and subsequently [itex][E_i,E_k]=0[/itex] at the point [itex]p[/itex] for all [itex]i,k[/itex]. As [itex][E_i,E_k]=\sum_j c^j_{ik}E_j[/itex] the structure functions [itex]c^j_{ik}[/itex] also vanish at [itex]p[/itex].

Now I compute [tex]\nabla_{E_k}[E_i,E_k]=[E_k,[E_i,E_k]]+\nabla_{[E_i,E_k]}E_k=\sum_{jl}c^l_{ik}c^j_{lk}E_lE_j+\nabla_{[E_i,E_k]}E_k=0[/tex] at [itex]p[/itex], where the first summand vanishes because of [itex]c^j_{ik}=0[/itex] and the second summand vanishes because the covariant derivative [itex]\nabla_XY[/itex] is tensorial in [itex]X[/itex] so it vanishes if [itex]X=0[/itex].

In total that means that in an normal orthonormal frame about a point [itex]p[/itex] not only all the covariant derivatives and the Lie-Bracket of the basis vectors vanish at [itex]p[/itex], but also the covariant derivative of the Lie-Bracket. Does that make sense? Or am I mistaken here?

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# Covariant derivative of Lie-Bracket in normal orthonormal frame

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