Consecutive shifts along orbits of vector fields

Abstract

Let $M$ be a smooth ($C^{\infty}$) manifold, $F_1,...,F_n$ be vector fields on $M$ generating the corresponding flows $Φ_1,...,Φ_n$, and $α_1,...,α_{n}:M\to \mathbb{R}$ smooth functions. Define the following map $f:M\to M$ by $$f(x)= Φ_n (... (Φ_2 (Φ_1 (x,α_1(x)), α_2(x)), ..., α_n(x)).$$ In this note we give a necessary and sufficient condition on vector fields $F_1,...,F_n$ and smooth functions $α_1,...,α_{n}$ 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.