论文标题
$ \ mathbb {e}^3 $中与平面场相关的一维叶子的主要周期
Principal cycles of one dimensional foliations associated to a plane field in $\mathbb{E}^3$
论文作者
论文摘要
在这项工作中，将分析与平面场$Δ_η$相关的一维单一叶子的$η$ - 原理周期（紧凑叶叶），由单位和普通矢量字段$η$ n $ \ m athbb e^3 $定义。叶子与$η$的轨道是正交的，是与平面场$Δ_η$的极端正常曲率方向相对应的积分曲线。 ％均表明，通常，给定$η$ - 首席周期，它可以使双曲线（Poincaré地图的首次返回的衍生物都通过对矢量场$η$的小变形而与单位循环的所有特征值脱节）。还表明，对于一组密集的单位矢量场，惠特尼（Whitney）的$ c^r $ $ $ - $η$ - 主体周期是双曲线。
In this work it will be analyzed $η$-principal cycles (compact leaves) of one dimensional singular foliations associated to a plane field $Δ_η$ defined by a unit and normal vector field $η$ in $ \mathbb E^3$. The leaves are orthogonal to the orbits of $η$ and are the integral curves corresponding to directions of extreme normal curvature of the plane field $Δ_η$. % It is shown that, generically, given a $η$-principal cycle it can be make hyperbolic (the derivative of the first return of the Poincaré map has all eigenvalues disjoint from the unit circle) by a small deformation of the vector field $η$. Also is shown that for a dense set of unit vector fields, with the weak $C^r$-topology of Whitney, the $η$-principal cycles are hyperbolic.
