arXiv 1807.045332018-07-12math.DS

曲面上向量场的准奇性

Primary singularities of vector fields on surfaces

Morris W. Hirsch, Francisco-Javier Turiel

摘要：

除非另有说明，否则在$C^\infty$范畴中工作，流形有空边界。设$X$和$Y$是流形$M$上的向量场.如果$[Y,X]=fX$对于某个连续函数$f\colon M\rightarrow\mathbb R$是$Y$跟踪$X$。零集${\mathsf Z}(X)$的子集$K$是$X$的本质块，如果它在${\mathsf Z}(X)$中是非空的、紧的、开的，并且它的Poincaré-Hopf指标不为零.有人说，如果$X$在$p$的$\infty$ -喷流是非平凡的，则$X$在$p$是非平坦的。如果关于$p$和跟踪$X$定义的任何向量场在$p$处为零，则称${\mathsf Z}(X)$的点$p$为$X$的主奇点。这是我们的主要结果：考虑定义在曲面$M$上的向量场$X$的本质块$K$。假设$X$在$K$的每个点处都是非平坦的。则$K$包含$X$的主奇点。因此，如果$M$是具有非零特征的紧致曲面，并且$X$无处平坦，则存在$X$的主奇异性。

英文摘要：

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 Poincaré-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 $X$.

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