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**Lines of principal curvature for mappings with Whitney umbrella singularities.**
*(English)*
Zbl 0611.53007

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.

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.

Reviewer: H.Guggenheimer

### MSC:

53A05 | Surfaces in Euclidean and related spaces |

37C75 | Stability theory for smooth dynamical systems |

### Keywords:

Whitney umbrella singularity; curvature lines; principally stable; Darboux umbilic; principal cycle; structurally stable
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\textit{C. Gutierrez} and \textit{J. Sotomayor}, Tôhoku Math. J. (2) 38, No. 1--2, 551--559 (1986; Zbl 0611.53007)

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### References:

[1] | C. GUTIERREZ AND J. SOTOMAYOR, Structurally stable configurationsof lines of princi-pal curvature, Asterisque 98-99 (1982), 195-215. · Zbl 0521.53003 |

[2] | C. GUTIERREZ AND J. SOTOMAYOR, An approximation theorem for immersions wit stable configurations of lines of principal curvature, Lectures Notes in Math. 1007, Springer Verlag, New York, 1983, 332-368. · Zbl 0528.53002 |

[3] | C. GUTIERREZ AND J. SOTOMAYOR, Principal lines on surfaces immersed with constan mean curvature, Trans. Amer. Math. Soc. 293 (1983), 751-766. JSTOR: · Zbl 0598.53007 |

[4] | N. KUIPER, Stable surfaces in Euclidean three space, Math. Scand. 36 (1975), 83-96 · Zbl 0298.57017 |

[5] | H. LEVINE AND R. THOM, Singularities of differentiate mappings, Reprinted i Lecture Notes in Math. 192, Springer Verlag, New York, 1971, 1-96. · Zbl 0216.45803 |

[6] | H. WHITNEY, The general type of singularity of a set of 2n–l smooth functions of variables, Duke Math. J. 10 (1943), 161-172. · Zbl 0061.37207 |

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