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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

Keywords:

Poincaré map

Citations:

Zbl 0266.34046
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References:

[1] Gutierrez, C.; Sotomayor, J.: Structurally stable configurations of lines of principal curvature. Asterisque 98-99 (1982). · Zbl 0521.53003
[2] Gutierrez, C.; Sotomayor, J.: An approximation theorem for immersions with stable configurations of lines of principal curvature. In: Geometric Dynamics. Proceedings. Lectures Notes in Math. 1007, Springer, (1983). · Zbl 0528.53002
[3] Gutierrez, C.; Sotomayor, J.: Closed principal lines and bifurcations. Bol. Soc. Brasil. Mat. 17 (1986). · Zbl 0608.53003
[4] Gutierrez, C.; Sotomayor, J.: Principal lines on surfaces immersed with constant mean curvature. Trans. Amer. Math. Soc., 239 (1986). · Zbl 0611.53007
[5] Gutierrez, C.; Sotomayor, J.: Lines of Curvature and Umbilic Points on Surfaces. In: 18 th Brasilian Math. Colloquium. IMPA, 1991. · Zbl 1160.53304
[6] Garcia, R.; Sotomayor, J.: Lines of Curvature Near Hyperbolic Principal Cycles. 3.IInt. Conf. on Dynamical Systems, Chile, 1990. Preprint IMPA, 1991. · Zbl 0786.53037
[7] Golubitsky, M.; Schaeffer, D.: Singularities and Groups in Bifurcation Theory. Vol. 1. Applied Math. Sciences 51, Springer Verlag, 1985. · Zbl 0607.35004
[8] Levine, H.; Thom, R.: Singularities of Differentiable Mappings. Lectures Notes in Math. 192, Springer, (1971). · Zbl 0216.45803
[9] Martinet, J.: Singularit?s des Functions et Applications Diff?rentiables. Monog. de Mat. Pont. Univ. Catol., Rio de Janeiro, N o 1, 1977. English Translation: Singularities of smooth functions and maps. London Math. Soc. Lecture Note Ser., 58, 1982.
[10] Sotomayor, J.: Closed lines of curvature on Weingarten immersions. Ann. Global Anal. Geom. 5 (1987). · Zbl 0638.53003
[11] Spivak, M.: A Comprehensive introduction to differential Geometry. Vol. 3. Publish or Perish, Berkeley 1979. · Zbl 0439.53003
[12] Struick, D.: Lectures on classical differential geometry. Addison Wesley, 1950. Reprinted by Dover, New York, 1988.
[13] Takens, F.: Normal forms for certain singularities of vector fields. Ann. Inst. Fourier 23 (1973), 2. · Zbl 0266.34046
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