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Zero sets of abelian Lie algebras of vector fields


Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X ∈ A. Theorem If K is a locally maximal compact set of zeroes of X ∈ A 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.

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