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Straightening theorem for vector fields

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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.