Straightening theorem for vector fields

In differential calculus, the domain-straightening theorem states that, given a vector field on a manifold, there exist local coordinates such that in a neighborhood of a point where is nonzero. The theorem is also known as straightening out of a vector field.

The Frobenius theorem in differential geometry can be considered as a higher-dimensional generalization of this theorem.

Proof

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It is clear that we only have to find such coordinates at 0 in  . First we write   where   is some coordinate system at  . Let  . By linear change of coordinates, we can assume   Let   be the solution of the initial value problem   and let

 

  (and thus  ) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that

 ,

and, since  , the differential   is the identity at  . Thus,   is a coordinate system at  . Finally, since  , we have:   and so   as required.

References

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  • Theorem B.7 in Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke. Poisson Structures, Springer, 2013.