EXISTENCE OF COMMON ZEROS FOR COMMUTING VECTOR FIELDS ON THREE MANIFOLDS

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 Poincare Hopf index of a vector field X in a region U. denoted by Ind(X, U); he asked: Given commuting vector fields X, Y and a region U where Ind(X, U) not equal 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 Cl vector fields X, Yon a 3-manifold, in any region U such that Ind(X, U) not equal 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.