{
  "id": "1606.08322",
  "version": "v1",
  "published": "2016-06-27T15:35:59.000Z",
  "updated": "2016-06-27T15:35:59.000Z",
  "title": "Zero sets of Lie algebras of analytic vector fields on real and complex 2-dimensional manifolds, II",
  "authors": [
    "Morris W. Hirsch",
    "F. -J. Turiel"
  ],
  "categories": [
    "math.DS"
  ],
  "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 $f\\colon M\\rightarrow\\mathbb F$. Let $K$ be a compact subset of the zero set ${\\mathsf Z}(X)$ such that ${\\mathsf Z}(X)-K$ is closed, with nonzero Poincar\\'e-Hopf index (for example $K={\\mathsf Z}(X)$ when $M$ is compact and $\\chi(M)\\neq 0$) 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-27T15:35:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F16",
      "58J20",
      "37F75",
      "37O25",
      "54H25"
    ],
    "keywords": [
      "analytic vector fields",
      "zero set",
      "nonzero poincare-hopf index",
      "finite-dimensional lie algebra",
      "nontrivial vector field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}