Primary singularities of vector fields on surfaces
| Autor: | Morris W. Hirsch, Francisco-Javier Turiel |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Surface (mathematics)
Jet (mathematics) General Mathematics Block (permutation group theory) Boundary (topology) Dynamical Systems (math.DS) 01 natural sciences Combinatorics Singularity 0103 physical sciences FOS: Mathematics Essential block Mathematics - Dynamical Systems 0101 mathematics Mathematics Continuous function (set theory) Zero set 010102 general mathematics Pure Mathematics Vector field Surface 010307 mathematical physics Geometry and Topology |
| Zdroj: | Geometriae Dedicata, vol 207, iss 1 |
| DOI: | 10.48550/arxiv.1807.04533 |
| Popis: | 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 $X$. |
| Databáze: | OpenAIRE |
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