Zero sets of Abelian Lie algebras of vector fields

TL;DR

This paper proves that on a 3D real manifold, if an abelian Lie algebra of analytic vector fields has a locally maximal compact zero set with nonzero Poincaré-Hopf index, then all vector fields in the algebra vanish at some point in that set.

Contribution

It establishes a new result linking zero sets of abelian Lie algebra vector fields with the Poincaré-Hopf index on 3D manifolds.

Findings

Nonzero Poincaré-Hopf index implies a common zero point for all vector fields in the algebra.

Locally maximal compact zero sets contain points where all vector fields vanish.

The result applies specifically to 3-dimensional real manifolds.

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.