Straightening theorem for vector fields explained

X on a manifold, there exist local coordinates y _{1,} ... , 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

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,} ... , f _{n)} . By linear change of coordinates, we can assume f( 0 ) = ( 1 , 0 , ... , 0 ). Let \Phi(t, p) be the solution of the initial value problem • x = f(x), x( 0 ) = p and let\psi(x _{1,} ... , x _{n)} = \Phi(x _{1,} ( 0 , x _{2,} ... , 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,} ... , x _{n)} = \Phi( 0 , ( 0 , x _{2,} ... , x _{n))} = ( 0 , x _{2,} ... , 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.