
Journal of Lie Theory 27 (2017), No. 4, 907914 Copyright Heldermann Verlag 2017 Zero Sets of Abelian Lie Algebras of Vector Fields Morris W. Hirsch Department of Mathematics, University of Wisconsin, 7926 Albe Road, Cross Plains, WI 53528, U.S.A. mwhirsch@chorus.net2 Assume M is a 3dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. Theorem. If K is a locally maximal compact set of zeroes of X and the PoincaréHopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish. Keywords: Analytic vector field, real manifold, abelian Lie algebra. MSC: 37C10, 37C35 [ Fulltextpdf (242 KB)] for subscribers only. 