Zero sets of Lie algebras of analytic vector fields on real and complex 2-dimensional manifolds, II

Abstract

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 $f:M\to \mathbb{F}$. 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 $\chi (M)\ne 0$) 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.