Lines of axial curvature on surfaces immersed in R4
Lines of axial curvature on surfaces immersed in R4
复制标题
浸没在 R4 中的表面上的轴向曲率线
DOI：
10.1016/s0926-2245(00)00015-2
复制
发表时间：
2000
影响因子：
0.5
通讯作者：
J. Sotomayor
中科院分区：
文献类型：
--
作者：
Ronaldo Garcia;J. Sotomayor
本文将传统上研究R3中曲面的脐线和主曲率线的主要概念推广到浸入R4中的曲面。从经典的第二基本形式和曲面在R4中的浸入的曲率椭圆出发，详细研究了a)轴脐点和b)轴曲率周期线的概念，这些点类似于经典脐点，因为在它们处曲率椭圆具有相等的轴线，在这里称为轴曲率周期，对应于经典R3情况下的主曲率周期和平均曲率周期。对于R4中的任何浸没表面，其轴向构形：主构形和平均曲率构形是相关联的。对于R3中的曲面，前者由脐线和主线简化为面形，后者由脐线和平均曲率线场的积分叶面给出面形。此外，对于R4中的曲面，将曲面浸入R3的主结构稳定性的概念推广到轴向结构稳定性的概念。给出了轴向结构稳定的充分条件，即轴向运动、轴向循环和所有其他轴曲率直线的渐近行为
The main notions concerning umbilics and lines of principal curvature, traditionally studied on surfaces in R3, are extended in this paper to surfaces immersed in R4. Departing from the classical second fundamental form and the ellipse of curvature of the immersion of a surface into R4here are studied in detail the concepts of a) axiumbilic points, analogous to classical umbilics since at them the ellipse of curvature has equal axes, and b) periodic lines of axial curvature, called here axial cycles, corresponding both to principal and mean curvature cycles in the classical R3case. To any immersed surface in R4its axial configurations: the principal configuration and the mean curvature configuration are associated. For surfaces in R3, the first one reduces to the configuration by umbilics and principal lines, while the second one gives the configuration by umbilics and integral foliations of the mean curvature line fields. Also the notion of principal structural stability of immersions of surfaces into R3is extended to that of axial structural stability, for the case of surfaces in R4. Sufficient conditions for the axial structural stability are provided in terms of axiumbilics, axial cycles and the asymptotic behavior of all the other lines of axial curvature