论文标题
关于$ {\ mathbb r}^4 $的高斯地图组件的奇异性
On singularities of the Gauss map components of surfaces in ${\mathbb R}^4$
论文作者
论文摘要
一般沉浸在$ \ mathbb r^4 $中的通用沉浸的高斯地图是沉浸式的。但是，这张地图以$ \ mathbb r^4 $的定向2牌的格拉马尼亚语为价值。由于该歧管具有两个球的产物结构，因此高斯图具有两个在球体上的值。我们研究高斯图组成部分的奇异性，并将其与通用浸入的几何特性相关联。此外，我们证明了奇异性通常是稳定的，并且我们将它们连接到表面的触点类型和$ \ Mathcal J $ -Holomorphic曲线相对于正交复合体结构$ \ MATHCAL J $ on $ \ MATHBB r^4 $。最后，我们得到了高斯河网类型的一些公式，涉及组件的奇异性的几何形状，并具有表面的几何形状和拓扑结构。
The Gauss map of a generic immersion of a smooth, oriented surface into $\mathbb R^4$ is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in $\mathbb R^4$. Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and $\mathcal J$-holomorphic curves with respect to an orthogonal complex structure $\mathcal J$ on $\mathbb R^4$. Finally, we get some formulas of Gauss-Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface.
