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**Closed principal lines and bifurcation.**
*(English)*
Zbl 0608.53003

The authors continue their series of investigations of lines of curvature on surfaces [the authors, Astérisque 98-99, 195-215 (1982; Zbl 0521.53003); Geometric dynamics, Proc. int. Symp., Rio de Janeiro/Brasil 1981, Lect. Notes Math. 1007, 332-368 (1983; Zbl 0528.53002); Dynamical systems, Proc. Colloq., Guanajuato/Mex. 1983, Aportaciones Mat., Notas Invest. 1, 115-126 (1985; Zbl 0587.58026), and Trans. Am. Math. Soc. 239, 751-766 (1986; Zbl 0598.53007)]. The main results deal with hyberbolic cycles (closed lines of curvature that do not pass through an umbilic and that behave like limit cycles in Poincaré-Bendixson theory). They consider the space of all \(C^ r\) immersions of a surface in \(R^ 3\) and prove a stability and a density theorem for cycles that in all immersions are hyperbolic with the possible exception of one semihyperbolic one, and the subset for which the semihyperbolic cycle is not the two-sided limit of umbilical separatrices or of any single principal curvature line. The proofs are explicit and complete.

Reviewer: H.Guggenheimer

### MSC:

53A05 | Surfaces in Euclidean and related spaces |

37C75 | Stability theory for smooth dynamical systems |

37C10 | Dynamics induced by flows and semiflows |

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\textit{C. Gutierrez} and \textit{J. Sotomayor}, Bol. Soc. Bras. Mat. 17, No. 1, 1--19 (1986; Zbl 0608.53003)

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### References:

[1] | [A-L] Andronov A; Leontovich E. et al.,Theory of Bifurcations of Dynamical Systems on the plane, John Wiley, New York, 1973. |

[2] | [Ch] Chuan-Chih Hsiung,A first course in differential geometry, John Wiley and Sons, N. Y. 1981. |

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[4] | [G-S.2] Gutierrez C., Sotomayor J.An approximation theorem for immersions with stable configurations of lines of principal curvature. Springer Lectures Notes in Math. 1007, (1983). · Zbl 0528.53002 |

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[8] | [G-S.6] Gutierrez C., Sotomayor J.,Bifurcations of principal lines connecting umbilical points. Preprint IMPA, 1986. |

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