An approximation theorem for immersions with stable configurations of lines of principal curvature
C Gutiérrez, J Sotomayor - … of the International Symposium held at the …, 2006 - Springer
C Gutiérrez, J Sotomayor
Geometric Dynamics: Proceedings of the International Symposium held at the …, 2006•SpringerIt is proved that every immersion of a compact oriented two-dimensional smooth manifold
into R 3 can be arbitrarily C 2-approximated by smooth immersions β whose principal
configurations P β=(U β, F β, f β) defined by umbilical points and families of lines of principal
curvature, are stable under C 3-sufficiently small perturbations of β. Actually, the elements β
are found in the class S r, r≥ 4, of C 3-principally structurally stable immersions, introduced
in [3]. Examples of immersions with recurrent lines of principal curvature are also given.
into R 3 can be arbitrarily C 2-approximated by smooth immersions β whose principal
configurations P β=(U β, F β, f β) defined by umbilical points and families of lines of principal
curvature, are stable under C 3-sufficiently small perturbations of β. Actually, the elements β
are found in the class S r, r≥ 4, of C 3-principally structurally stable immersions, introduced
in [3]. Examples of immersions with recurrent lines of principal curvature are also given.
Abstract
It is proved that every immersion of a compact oriented two-dimensional smooth manifold into R3 can be arbitrarily C2-approximated by smooth immersions β whose principal configurations Pβ = (Uβ ,Fβ ,fβ) defined by umbilical points and families of lines of principal curvature, are stable under C3-sufficiently small perturbations of β. Actually, the elements β are found in the class S r, r≥4, of C3-principally structurally stable immersions, introduced in [3].
Examples of immersions with recurrent lines of principal curvature are also given.
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