沿向量场轨道的连续位移
Consecutive shifts along orbits of vector fields
摘要：
设$M$是一个光滑（$C^{\infty}$）流形，$F_1,...,F_n$是$M$上生成相应流$Φ_1,...,Φ_n$的向量场，$α_1,...,α_{n}:M\to \mathbb{R}$是光滑函数.本文给出了向量场$f:M\to M$和光滑函数$$f(x)= Φ_n (... (Φ_2 (Φ_1 (x,α_1(x)), α_2(x)), ..., α_n(x)).$$使$f$为局部单同态的一个充要条件。证明了这个条件对于相应的向量场和函数的同时置换是不变的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
