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


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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