Journal of Lie Theory 27 (2017), No. 4, 907--914
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.
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.
Theorem. If K is a locally maximal compact set of zeroes of X and the Poincaré-Hopf 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
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