Bifurcations of Umbilic Points and Related Principal Cycles

Surfaces with one constant principal curvature in three-dimensional space forms

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal plane. In some cases, a kind of trichotomy phenomenon is observed: the curves of the foliations may be circles, hyperbolas or horocycles, depending of whether the constant principal curvature is respectively larger, smaller of equal to one (not necessarily in this order). We descr…

解析曲面的退化脐点

Degenerate Umbilic Points of Analytic Surfaces

本文通过更简单的几何论证证明解析曲面上脐点指数不能大于1，并基于退化性建立指数下界，为Carathéodory猜想提供新见解。

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 alo…

Affine curvature lines of surfaces in 3-space

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines in the neighborhood of points with double eigenvalues (but not umbilics) of the affine shape operator and parabolic points

Axiumbilic Singular Points on Surfaces Immersed in R4 and their Generic Bifurcations

Here are described the axiumbilic points that appear in generic one parameter families of surfaces immersed in R4. At these points the ellipse of curvature of the immersion, Little, Garcia - Sotomayor has equal axes. A review is made on the basic preliminaries on axial curvature lines and the associated axiumbilic points which are the singularities of the fields of principal, mean axial lines}, axial crossings and the quartic differential equation defining them. The Lie-Cartan vector field suspension of the quartic differential equation, giving a line field tangent to the Lie-Cartan surface (i…

Bifurcations of robust features on surfaces in the Minkowski 3-space

We obtain the bifurcation of some special curves on generic 1-parameter families of surfaces in the Minkowski 3-space. The curves treated here are the locus of points where the induced pseudo metric is degenerate, the discriminant of the lines principal curvature, the parabolic curve and the locus of points where the mean curvature vanishes.

Bifurcation of Fredholm Maps II; The Dimension of the Set of Bifurcation Points

We obtain an estimate for the covering dimension of the set of bifurcation points for solutions of nonlinear elliptic boundary value problems from the principal symbol of the linearization of the problem along the trivial branch of solutions.

Admissible surfaces in progressive addition lenses

Progressive addition lenses contain a surface of spatially-varying curvature, which provides variable optical power for different viewing areas over the lens. We derive complete compatibility equations that provide the exact magnitude of cylinder along lines of curvatures on any arbitrary PAL smooth surface. These equations reveal that, contrary to current knowledge, cylinder, and its derivative, does not only depend on principal curvature and its derivatives along the principal line but also on the geodesic curvature and its derivatives along the line orthogonal to the principal line. We quan…

Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows

In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper we investigate conditions under which evolving a smooth rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A surprising variety of behaviours is uncovered: convergence to round spheres …

Lines of Mean Curvature on Surfaces Immersed in R3

Associated to oriented surfaces immersed in R^3 here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k_1, k_2 of the immersion, is positive. The leaves of the foliations are the lines of M- mean curvature, along which the normal curvature of the immersion is given by a function M = M(k_1,k_2) in [k_1, k_2], called a M- mean curvature, whose properties extend and unify those of the arithmetic H=(k_1+k_2)/2, the geometric K^{1/2} and harmonic K/H=((1/{k_1} + 1/{k_2})/2…

拉格朗日系统与测地线的分支现象 I

Bifurcations for Lagrangian systems and geodesics I

本文研究了具有不同边界条件和约束的拉格朗日系统中的分支现象，通过三种主要配置探讨了莫尔斯理论与多重解存在的关系，提出了分支条件的必要充分条件。