论文标题
可集成性和适应复杂的结构，以使平面矢量场在平面上平滑
Integrability and adapted complex structures to smooth vector fields on the plane
论文作者
论文摘要
Riemann表面上的单数复合分析矢量场具有多种几何特性（单数意味着可允许极点和基本奇异性）。我们描述了单数复杂分析矢量字段$ \ mathbb {x} $与光滑矢量字段$ x $之间的关系。我们的近似路线研究了实际平滑矢量场的三个可集成性概念$ x $与飞机或球体上的奇异性。第一个概念与Cauchy-Riemann方程有关，我们说，如果存在与此复杂结构提供的平面上的奇异复杂分析矢量字段$ x $，则vector field $ x $承认了一个改装的复杂结构$ j $，因此$ x $是$ x $的真正部分。 $ x $的第二个集成性概念是存在第一个积分$ f $，它平稳，并且在$ x $的奇点之外具有非消失的差异。第三个概念是，$ x $在其奇点之外承认了全球流动箱图，即矢量字段$ x $是在差异性下的琐碎水平矢量场的提升。我们研究三个概念之间的关系。描述了三个集成性概念的拓扑障碍（本地和全球）。使用规范不变区域提供了单数复合物分析矢量场X的构建。
Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields $\mathbb{X}$ and smooth vector fields $X$. Our approximation route studies three integrability notions for real smooth vector fields $X$ with singularities on the plane or the sphere. The first notion is related to Cauchy-Riemann equations, we say that a vector field $X$ admits an adapted complex structure $J$ if there exists a singular complex analytic vector field $X$ on the plane provided with this complex structure, such that $X$ is the real part of $\mathbb{X}$. The second integrability notion for $X$ is the existence of a first integral $f$, smooth and having non vanishing differential outside of the singularities of $X$. A third concept is that $X$ admits a global flow box map outside of its singularities, i.e. the vector field $X$ is a lift of the trivial horizontal vector field, under a diffeomorphism. We study the relation between the three notions. Topological obstructions (local and global) to the three integrability notions are described. A construction of singular complex analytic vector fields X using canonical invariant regions is provided.
