Lines of curvature near principal cycles (Q1202532)

If a surface \(\alpha\) admits a closed line of curvature of multiplicity \(n\), the authors show that there exist local coordinates following \textit{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.