Lines of principal curvature around umbilics and Whitney umbrellas

Abstract

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