A PROJECTIVE INVARIANT FOR SWALLOWTAILS AND GODRONS, AND GLOBAL THEOREMS ON THE FLECNODAL CURVE
A PROJECTIVE INVARIANT FOR SWALLOWTAILS AND GODRONS, AND GLOBAL THEOREMS ON THE FLECNODAL CURVE
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DOI：
10.17323/1609-4514-2006-6-4-731-768
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发表时间：
2006-01-01
影响因子：
0.8
通讯作者：
Uribe-Vargas, Ricardo
中科院分区：
文献类型：
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作者：
Uribe-Vargas, Ricardo
我们展示了一些通用的（鲁棒的）性质的光滑表面沉浸在真实的3-空间（欧几里德，仿射或投影），在附近的一个godron（也称为尖点高斯）：一个孤立的抛物点在（唯一的）渐近方向是相切的抛物曲线。借助这些性质和一个射影不变量，我们关联到每个godron，我们提出了所有可能的局部配置的切平面，自交线，尖形边缘和flecnodal曲线在R-3中的一个通用燕尾。我们提出了一些全球性的结果，例如：在一个双曲盘的一般光滑曲面，飞节曲线有奇数个横向自相交（因此至少有一个自相交）。
We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also cusp of Gauss): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we present all possible local configurations of the tangent plane, the self-intersection line, the cuspidal edge and the flecnodal curve at a generic swallowtail in R-3. We present some global results, for instance: In a hyperbolic disc of a generic smooth surface, the,flee-nodal curve has an odd number of transverse self-intersections (hence at least One self-intersection).