## 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.

### MSC:

 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:

### References:

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