Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of common zeros for commuting vector fields on $3$-manifolds   In $64$ E. Lima proved that commuting vector fields on surfaces with non-zero
Euler characteristic have common zeros. Such statement is empty in dimension
$3$, since all the Euler characteristics vanish. Nevertheless,
\cite{Bonatti_analiticos} proposed a local version, replacing the Euler
characteristic by the Poincar\'e-Hopf index of a vector field $X$ in a region
$U$, denoted by $\operatorname{Ind}(X,U)$; he asked:
  \emph{Given commuting vector fields $X,Y$ and a region $U$ where
$\operatorname{Ind}(X,U)\neq 0$, does $U$ contain a common zero of $X$ and
$Y$?}
  \cite{Bonatti_analiticos} gave a positive answer in the case where $X$ and
$Y$ are real analytic.
  In this paper, we prove the existence of common zeros for commuting $C^1$
vector fields $X$, $Y$ on a $3$-manifold, in any region $U$ such that
$\operatorname{Ind}(X,U)\neq 0$, assuming that the set of collinearity of $X$
and $Y$ is contained in a smooth surface. This is a strong indication that the
results in \cite{Bonatti_analiticos} should hold for $C^1$-vector fields.

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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con