A mapping of a 2-manifold \(\alpha\) : \(M\to {\mathbb{R}}^ 3\) has a Whitney umbrella singularity at a point p if there is a chart for which \(\alpha_ v(0)=0\) and \(\alpha_ u(0)\), \(\alpha_{uu}(0)\), \(\alpha_{vv}(0)\) span \({\mathbb{R}}^ 3\). The authors show that a Whitney umbrella has a single tangent to all curvature lines. Other results obtained by references to previous papers [the authors, Astérisque 98- 99, 195-215 (1983; Zbl 0521.53003), Lect. Notes Math. 1007, 332-368 (1983; Zbl 0528.53002)] are: 1. Mappings of M are \(C^ 2\) principally stable as long as no singularities other than Whitney umbrella appear. 2. Let \(S^ r(M)\) be the set of maps \(\alpha\) such that (a) every umbilic is a (nondegenerate) Darboux umbilic and all singularities are Whitney umbrellas, (b) any principal cycle is a hyperbolic cycle of the foliation to which it belongs, (c) the limit set of every principal line is the union of singular points, umbilics, and principal cycles, (d) no umbilical separatrix is separatrix for 2 distinct umbilics nor counted double for one umbilic. Then for \(r\geq 4\), \(S^ r\) is open in the set of \(C^ 3\) maps of M, every \(\alpha\) in it is principally structurally stable, and \(S^ r\) is dense in the set of \(C^ 2\) maps of M.