Zero sets of Abelian Lie algebras of vector fields

Abstract

Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. The following result is proved: If K is a locally maximal compact set of zeroes of X and the Poincar'e-Hopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish.

8 pages

References in corpus (3)

• Fixed points of local actions of nilpotent Lie groups on surfaces
• A generalized Poincaré-Hopf index theorem
• Existence of common zeros for commuting vector fields on $3$-manifolds