2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103858Let $M$ 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.9 pages, submitted to the Proceedings of the conference FOLIATIONS-2005, Poland, LodzDifferential GeometryDynamical Systems57R30text