Existence of common zeros for commuting vector fields on three manifolds

Abstract : In 1964, 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, C. Bonatti proposed in 1992 a local version, replacing the Euler characteristic by the Poincaré–Hopf index of a vector field $X$ in a region $U$, denoted by $ \operatorname{Ind}(X,U)$; he asked: 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 $Y$? A positive answer was given in the case where $X$ and $Y$ are real analytic, in the same article where the above question was posed. 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 for analytic vector fields should hold in the $C^1$ setting.
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Contributor : Imb - Université de Bourgogne <>
Submitted on : Tuesday, December 12, 2017 - 1:30:34 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

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Christian Bonatti, Bruno Santiago. Existence of common zeros for commuting vector fields on three manifolds. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2017, 67 (4), pp.1741 - 1781. ⟨10.5802/aif.3121⟩. ⟨hal-01661895⟩

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