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arXiv · 1108.2643

The Structure of Stable Vector Fields on Surfaces

John Berman·Sergei Bernstein
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Abstract

The Poincare-Hopf theorem tells us that given a smooth, structurally stable vector field on a surface of genus g, the number of saddles is 2-2g less than the number of sinks and sources. We generalize this result by introducing a more complex combinatorial invariant. Using this tool, we demonstrate that many such structurally stable vector fields are equivalent up to a set of basic operations. We show in particular that for the sphere, all such vector fields are equivalent.

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John Berman, Sergei Bernstein. 2011-08-12. The Structure of Stable Vector Fields on Surfaces. https://arxiv.org/abs/1108.2643

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