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-Poincaré characteristic in terms of the number of umbilics and umbrellas.
Tohoku Math. J. (2)
52(2):
163-172
(2000).
DOI: 10.2748/tmj/1178224605