浸在${\mathbb R} ^4$中的表面上的主中曲率叶理
Principal Mean Curvature Foliations on Surfaces immersed in ${\mathbb R} ^4$
摘要：
本文研究了浸入${\mathbb R}^4$中的曲面上沿平均法曲率向量方向极端弯曲的曲线族--叶理的定性性质。典型的奇异性和周期的描述，提供了充分条件，可能也是必要的，为结构稳定性性的配置，这样的叶理和他们的奇异性，根据小$C^3$扰动的浸没。这些条件表示为正规奇异性和脐奇异性的Darbouxian型、圈的双曲性、奇异分离线和叶理的其他典型曲线的渐近行为。它们推广了Gutierrez和Sotomayor在1982年给出的关于浸在${\mathbb R}^3$中的表面的主叶理和脐点的结果。达布仙条件和双曲性，可计算的衍生物的浸入在奇点和周期的表达式，提供。最后指出了目前${\mathbb R}^3$到${\mathbb R}^4$的扩展与其它相关扩展的联系以及本文中尚待解决的问题。
英文摘要：
Here are studied qualitative properties of the families of curves --foliations-- on a surface immersed in ${\mathbb R}^4$, along which it bends extremally in the direction of the mean normal curvature vector. Typical singularities and cycles are described, which provide sufficient conditions, likely to be also necessary, for the structural stability of the configuration of such foliations and their singularities, under small $C^3$ perturbations of the immersion. The conditions are expressed in terms of Darbouxian type of the normal and umbilic singularities, the hyperbolicity of cycles, and the asymptotic behavior of singularity separatrices and other typical curves of the foliations. They extend those given by Gutierrez and Sotomayor in 1982 for principal foliations and umbilic points of surfaces immersed in ${\mathbb R}^3$. Expressions for the Darbouxian conditions and for the hyperbolicity, calculable in terms of the derivatives of the immersion at singularities and cycles, are provided. The connection of the present extension from ${\mathbb R}^3$ to ${\mathbb R}^4$to other pertinent ones as well as some problems left open in this paper are proposed at the end.
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