Frobenius' Theorem

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I need to understand a certain characterization of Frobenius' Theorem, part of which contains the following statement:

[tex]\nabla_{[a}\xi_{b]}=\xi_{[a}v_{b]}[/tex] for some dual vector field [tex]v_{b}[/tex] if and only if [tex]\xi_{[a}\nabla_{b}\xi_{c]}=0[/tex], where [tex]\xi^a\xi_a\neq 0[/tex].

Is it obvious, or difficult to prove? I do not see the converse ...
 
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I need to understand a certain characterization of Frobenius' Theorem, part of which contains the following statement:

[tex]\nabla_{[a}\xi_{b]}=\xi_{[a}v_{b]}[/tex] for some dual vector field [tex]v_{b}[/tex] if and only if [tex]\xi_{[a}\nabla_{b}\xi_{c]}=0[/tex], where [tex]\xi^a\xi_a\neq 0[/tex].

Is it obvious, or difficult to prove? I do not see the converse ...

Ι wouldn't say it is hard, it is just -as many proofs in geometry- rather long and tedious. For the converse part, the condition [tex]\xi_{[a}\nabla_{b}\xi_{c]}=0[/tex] is the integrability condition that guarantees us that the partial differential equation [tex]\nabla_{[a}\xi_{b]}=\xi_{[a}v_{b]}[/tex] is solvable for [tex]v[/tex].
 

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