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The following claim can be found in page 27 of Petersen's book "Riemannian Geometry":

"... any three vectors can be extended to vector fields that commute."

I have been unable to prove the statement, mostly because the equations determining the commutativity of vector fields are a system of nonlinear partial differential equations, although their symmetry might make them easier to treat.

I was wondering if someone could help me find an existence theorem for the Cauchy problem of such equations, or figure out a way to prove such a claim.

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  • $\begingroup$ On $\Bbb R^n$, this is clearly true, since you can just let each vector field be constant. That means you can do it on a coordinate neighbourhood. What to do after that, I don't know. Could you do it in $\Bbb R^n$ in such a way that the fields all have compact support, perhaps? $\endgroup$ – Arthur Oct 3 '17 at 11:14
  • $\begingroup$ That's exactly what I've been thinking, but have not been able to make them compactly supported while maintaining the commutativity. $\endgroup$ – Ignatius Oct 3 '17 at 11:19
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    $\begingroup$ The trick to get compact support is to use a pushforward instead of a cutoff function: let $B$ be a geodesic ball, choose a radial diffeomorphism $f : \mathbb R^n \to B$ and take the vector fields $f_* X_i$, where $X_i$ are the appropriate parallel vector fields on $\mathbb R^n$. By choosing $f$ with the correct asymptotic you can ensure that these vector fields can be smoothly extended by zero outside $B$. $\endgroup$ – Anthony Carapetis Oct 3 '17 at 13:03
  • $\begingroup$ The last problem here has a more thorough sketch. $\endgroup$ – Anthony Carapetis Oct 3 '17 at 13:09
  • $\begingroup$ That is a very good idea, indeed! $\endgroup$ – Ignatius Oct 7 '17 at 21:09

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