The Structure of Stable Vector Fields on Surfaces

TL;DR

This paper introduces a new combinatorial invariant to analyze structurally stable vector fields on surfaces, generalizing classical results and showing equivalence classes, especially on the sphere, through basic operations.

Contribution

It develops a novel combinatorial invariant for stable vector fields on surfaces and demonstrates their classification up to basic operations, extending classical theorems.

Findings

All stable vector fields on the sphere are equivalent under basic operations.

The new invariant generalizes the Poincare-Hopf theorem for complex surface structures.

Many stable vector fields can be classified into equivalence classes using this invariant.

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.