Primary singularities of vector fields on surfaces

TL;DR

This paper proves that on surfaces, essential blocks of vector fields with non-flat points necessarily contain primary singularities, highlighting fundamental properties of vector fields related to their zero sets and tracking behavior.

Contribution

It establishes the existence of primary singularities within essential blocks of vector fields on surfaces under non-flatness conditions, advancing understanding of vector field singularities.

Findings

Essential blocks with non-flat vector fields contain primary singularities.

On compact surfaces with non-zero characteristic, primary singularities always exist for nowhere flat vector fields.

The results connect tracking behavior and singularity structure of vector fields on surfaces.

Abstract

Unless another thing is stated one works in the $C^\infty$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb R$. A subset $K$ of the zero set ${\mathsf Z}(X)$ is an essential block for $X$ if it is non-empty, compact, open in ${\mathsf Z}(X)$ and its Poincar\'e-Hopf index does not vanishes. One says that $X$ is non-flat at $p$ if its $\infty$-jet at $p$ is non-trivial. A point $p$ of ${\mathsf Z}(X)$ is called a primary singularity of $X$ if any vector field defined about $p$ and tracking $X$ vanishes at $p$. This is our main result: Consider an essential block $K$ of a vector field $X$ defined on a surface $M$. Assume that $X$ is non-flat at every point of $K$. Then $K$ contains a primary singularity of $X$. As a consequence, if $M$ is a compact surface with non-zero characteristic and $X$ is nowhere flat, then there exists a primary singularity of $X$.