Lines of curvature near principal cycles. (English) Zbl 0768.53002

If a surface \(\alpha\) admits a closed line of curvature of multiplicity \(n\), the authors show that there exist local coordinates following F. Takens [Ann. Inst. Fourier 23, No. 2, 163-195 (1973; Zbl 0266.34046)] for which the curvature lines are given by \(du = 0\), \(dv-v^ n(a- bv^{n-1})du = 0\), \(a\), \(b\) depending on the \(2n+1\)-jet along \(u = 0\). They also prove for \(n \geq 2\), the curvature not constant for the curve \(u = 0\), that the Poincaré map (along \(u = 0\)) of a deformation \(\alpha + k'(u)\sum^{n-i}_ 1(\varepsilon_ iv^ i/i!)N_ \alpha(u)\) provides a universal unfolding for that deformation. The main tools are computations to express the Poincaré map at \(u = 0\) in terms of curvature dates of the surface and the curve.


53A05 Surfaces in Euclidean and related spaces
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory


Poincaré map


Zbl 0266.34046
Full Text: DOI


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