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 ${\Phi }_1,...,{\Phi }_n$, and ${\alpha }_1,...,{\alpha }_n:M\to \mathbb{R}$ smooth functions. Define the following map $f:M\to M$ by $$f(x)={\Phi }_n(...({\Phi }_2({\Phi }_1(x,{\alpha }_1(x)),{\alpha }_2(x)),...,{\alpha }_n(x)).$$ In this note we give a necessary and sufficient condition on vector fields $F_1,...,F_n$ and smooth functions ${\alpha }_1,...,{\alpha }_n$ 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, Lodz