Consecutive shifts along orbits of vector fields

الأصل · EN

Let M be a smooth (C∞) manifold, F₁,...,Fₙ be vector fields on M generating the corresponding flows Φ₁,...,Φₙ, and α₁,...,αₙ:M→ R smooth functions. Define the following map f:M→ M by f(x)= Φₙ (... (Φ₂ (Φ₁ (x,α₁(x)), α₂(x)),..., αₙ(x)). In this note we give a necessary and sufficient condition on vector fields F₁,...,Fₙ and smooth functions α₁,...,αₙ for f to be a local diffeomorphism. It turns out that this condition is invariant with respect to the simultaneous permutation of the corresponding vector fields and functions.