Codimension two Umbilic points on Surfaces Immersed in R^3

In this paper is studied the behavior of lines of curvature near umbilic points that appear generically on surfaces depending on two parameters.Comment: 19 pages, 10 figure

Geometric Mean Curvature Lines on Surfaces Immersed in R3

Here are studied pairs of transversal foliations with singularities, defined on the Elliptic region (where the Gaussian curvature $\mathcal K$ is positive) of an oriented surface immersed in $\mathbb R^3$. The leaves of the foliations are the lines of geometric mean curvature, along which the normal curvature is given by $\sqrt {\mathcal K}$, which is the geometric mean curvature of the principal curvatures $k_1, k_2$ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves}, where $k_1 = k_2$ and ${\mathcal K} = 0$, respectively. Here are determined the structurally stable patterns of geometric mean curvature lines near the umbilic points, parabolic curves and geometric mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established. This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Geometric Mean Curvature Structural Stability. This study, outlined at the end of the paper, is a natural analog and complement for the Arithmetic Mean Curvature and Asymptotic Structural Stability of immersed surfaces studied previously by the authors.Comment: 21 pages, 5 figures. To appear in Annales de la Faculte de Sciences de Toulous

On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points

In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.Comment: 12 pages, 5 figure

Lines of axial curvature at critical points on surfaces mapped into R4

In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter families of mappings of surfaces into R4. As the parameter crosses a critical bifurcation value, at which the mapping has a critical point, it is described how the axial umbilic points, which are the singularities of the axial curvature configurations at regular points, move along smooth arcs to reach the critical point. The numbers of such arcs and their axial umbilic types are fully described for a typical family of mappings with a critical point.Comment: 19 pages, 12 figure