Straightening theorem for vector fields

In differential calculus, the domain-straightening theorem states that, given a vector field [math]\displaystyle{ X }[/math] on a manifold, there exist local coordinates [math]\displaystyle{ y_1, \dots, y_n }[/math] such that [math]\displaystyle{ X = \partial / \partial y_1 }[/math] in a neighborhood of a point where [math]\displaystyle{ X }[/math] 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

It is clear that we only have to find such coordinates at 0 in [math]\displaystyle{ \mathbb{R}^n }[/math]. First we write [math]\displaystyle{ X = \sum_j f_j(x) {\partial \over \partial x_j} }[/math] where [math]\displaystyle{ x }[/math] is some coordinate system at [math]\displaystyle{ 0 }[/math]. Let [math]\displaystyle{ f = (f_1, \dots, f_n) }[/math]. By linear change of coordinates, we can assume [math]\displaystyle{ f(0) = (1, 0, \dots, 0). }[/math] Let [math]\displaystyle{ \Phi(t, p) }[/math] be the solution of the initial value problem [math]\displaystyle{ \dot x = f(x), x(0) = p }[/math] and let

[math]\displaystyle{ \psi(x_1, \dots, x_n) = \Phi(x_1, (0, x_2, \dots, x_n)). }[/math]

[math]\displaystyle{ \Phi }[/math] (and thus [math]\displaystyle{ \psi }[/math]) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that

[math]\displaystyle{ {\partial \over \partial x_1} \psi(x) = f(\psi(x)) }[/math],

and, since [math]\displaystyle{ \psi(0, x_2, \dots, x_n) = \Phi(0, (0, x_2, \dots, x_n)) = (0, x_2, \dots, x_n) }[/math], the differential [math]\displaystyle{ d\psi }[/math] is the identity at [math]\displaystyle{ 0 }[/math]. Thus, [math]\displaystyle{ y = \psi^{-1}(x) }[/math] is a coordinate system at [math]\displaystyle{ 0 }[/math]. Finally, since [math]\displaystyle{ x = \psi(y) }[/math], we have: [math]\displaystyle{ {\partial x_j \over \partial y_1} = f_j(\psi(y)) = f_j(x) }[/math] and so [math]\displaystyle{ {\partial \over \partial y_1} = X }[/math] as required.

References

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