LINES OF PRINCIPAL CURVATURE AROUND UMBILICS AND WHITNEY UMBRELLAS
LINES OF PRINCIPAL CURVATURE AROUND UMBILICS AND WHITNEY UMBRELLAS
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脐带和惠特尼伞周围的主曲率线
DOI：
10.2748/tmj/1178224605
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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;C. Gutierrez;J. Sotomayor
本文研究了曲面到R3的映射的唯一稳定奇点Whitney伞附近的曲率线的构形。这种配置的模式是建立和特点的3喷的地图。结果是用来建立一个表达式的欧拉-庞加莱特性的脐和伞的数量。1.导论.光滑映射a：M ->> R3的弯曲或曲率模式，其中M是紧致定向二维流形，在这里将由奇点Sa表示，在奇点Sa处映射具有小于2的秩并且弯曲可以被认为是无限的;脐点Ua，在脐点Ua处弯曲是有限的但是在所有方向上相等：以及由主曲率线族T1，a和Ti，a定义在M \（Ua U 4.他们表明，Darbouxian脐点的特点是那些具有局部结构稳定的配置，在小的C3变形的表面。另见布鲁斯和菲达尔的著作（B-F）。Gutierrez和Sotomayor（GS 2）对奇点集Sa附近的主叶理进行了研究，旨在表征其局部稳定性。为此，他们给出了两个充分条件，其中第一个是在奇点理论（G-G）意义下映射稳定的Whitney奇点条件。然而，在本文中，一个错误的
In this paper is studied the configuration of lines of curvature near a Whitney umbrella which is the unique stable singularity for maps of surfaces into R 3. The pattern of such configuration is established and characterized in terms of the 3-jet of the map. The result is used to establish an expression for the Euler-Poincare characteristic in terms of the number of umbilics and umbrellas. 1. Introduction. The bending or curvature pattern of a smooth mapping a : M -» R3, where M is a compact oriented two dimensional manifold, will be represented here by singular points, Sa, at which the mapping has rank less than 2 and the bending can be re- garded to be infinite; the umbilic points, Ua, at which the bending is finite but equal in all directions: and by the family of lines of principal curvature T\,a and Ti,a defined on M \ (Ua U 4. They showed that Darbouxian umbilic points characterize those with local structurally stable configuration, under small C3 deformations of the surface. See also the work of Bruce and Fidal (B-F). A study of principal foliations near the set Sa of singular points, aiming to characterize their local stability, was carried out by Gutierrez and Sotomayor (GS2). To this end they gave two sufficient conditions, the first of which is the Whitney singularity condition for stability of mappings in the sense of Singularity Theory (G-G). However, in this paper, an erroneous