Straightening theorem for vector fields

In differential calculus, the domain-straightening theorem states that, given a vector field $X$ on a manifold, there exist local coordinates $y_{1},\dots ,y_{n}$ such that $X=\partial /\partial y_{1}$ in a neighborhood of a point where $X$ 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 $\mathbb {R} ^{n}$ . First we write $X=\sum _{j}f_{j}(x){\partial \over \partial x_{j}}$ where $x$ is some coordinate system at $0$ . Let $f=(f_{1},\dots ,f_{n})$ . By linear change of coordinates, we can assume $f(0)=(1,0,\dots ,0).$ Let $\Phi (t,p)$ be the solution of the initial value problem ${\dot {x}}=f(x),x(0)=p$ and let

\psi (x_{1},\dots ,x_{n})=\Phi (x_{1},(0,x_{2},\dots ,x_{n})).

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

{\partial  \over \partial x_{1}}\psi (x)=f(\psi (x))

and, since $\psi (0,x_{2},\dots ,x_{n})=\Phi (0,(0,x_{2},\dots ,x_{n}))=(0,x_{2},\dots ,x_{n})$ , the differential $d\psi$ is the identity at $0$ . Thus, $y=\psi ^{-1}(x)$ is a coordinate system at $0$ . Finally, since $x=\psi (y)$ , we have: ${\partial x_{j} \over \partial y_{1}}=f_{j}(\psi (y))=f_{j}(x)$ and so ${\partial \over \partial y_{1}}=X$ as required.

References

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

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