实与复二维流形上解析向量场李代数的零集，II
Zero sets of Lie algebras of analytic vector fields on real and complex 2-dimensional manifolds, II
摘要：
在实（$\mathbb F=\mathbb R$）或复（$\mathbb F=\mathbb C$）解析连通2-流形$M$上，考虑两个向量场$X,Y$ .我们说$Y$ { \it跟踪} $X$，如果$[Y,X]=fX$对于某个连续函数$f\colon M\rightarrow\mathbb F$。令$K$是零集${\mathsf Z}(X)$的紧致子集，使得${\mathsf Z}(X)-K$是闭的，具有非零的Poincaré-Hopf指标（例如$K={\mathsf Z}(X)$，当$M$是紧致的，而$χ(M)\neq 0$），令$\mathcal G$是$M$上解析向量场的有限维李代数。PDMATHBNB {PDMATHBNC定理。}让$X$是解析的和非平凡的。如果$\mathcal G$的每个元素都跟踪$X$，并且在${\mathsf i}_K (X)$为正且甚至没有$\mathcal G$的商同构于${\mathfrak {s}}{\mathfrak {l}} (2,\mathbb C)$的复杂情况下，则$\mathcal G$在$K$中有某个零。PDMATHBNB {PDMATHBNC推论。}如果$Y$跟踪一个非平凡向量场$X$，它们都是解析的，那么$Y$在$K$中的某个地方消失。此外，还证明了某些类型的变换群的不动点定理。给出了几个说明性的例子。
英文摘要：
On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb F$. Let $K$ be a compact subset of the zero set ${\mathsf Z}(X)$ such that ${\mathsf Z}(X)-K$ is closed, with nonzero Poincaré-Hopf index (for example $K={\mathsf Z}(X)$ when $M$ is compact and $χ(M)\neq 0$) and let $\mathcal G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$. \smallskip {\bf Theorem.} Let $X$ be analytic and nontrivial. If every element of $\mathcal G$ tracks $X$ and, in the complex case when ${\mathsf i}_K (X)$ is positive and even no quotient of $\mathcal G$ is isomorphic to ${\mathfrak {s}}{\mathfrak {l}} (2,\mathbb C)$, then $\mathcal G$ has some zero in $K$. \smallskip {\bf Corollary.} If $Y$ tracks a nontrivial vector field $X$, both of them analytic, then $Y$ vanishes somewhere in $K$. \smallskip Besides fixed point theorems for certain types of transformation groups are proved. Several illustrative examples are given.
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