Existence of common zeros for commuting vector fields on $3$-manifolds

Abstract

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 Poincaré-Hopf index of a vector field $X$ in a region $U$, denoted by $\mathrm{Ind}(X,U)$; he asked: \emph{Given commuting vector fields $X,Y$ and a region $U$ where $\mathrm{Ind}(X,U)\ne 0$, 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 $\mathrm{Ind}(X,U)\ne 0$, 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.

Final version, to appear in Annales de L'Institut Fourier

Cited by in corpus (2)

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