Bifurcations of Umbilic Points and Related Principal Cycles

Abstract

The simplest patterns of qualitative changes on the configurations of lines of principal curvature} around umbilic points on surfaces whose immersions into ${\mathbb{R}}^3$ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper. Global effects, due to umbilic bifurcations, on these configurations such as the appearance and annihilation of periodic principal lines, called also principal cycles, are also studied here.

26 pages, 12 figures

References in corpus (2)

• Harmonic Mean Curvature Lines on Surfaces Immersed in R3
• Geometric Mean Curvature Lines on Surfaces Immersed in R3

Cited by in corpus (6)

• Lines of Curvature on Surfaces, Historical Comments and Recent Developments
• Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in ${\mathbb{R}}^3$
• Lines of Principal Curvature near Singular End Points of Surfaces in R3
• On the multiplicity of umbilic points
• On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points
• Principal configurations around umbilics of spacelike surfaces in null hypersurfaces of ${\mathbb{R}}_1^4$