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.