浸入R4的曲面上的公理奇点及其一般分支
Axiumbilic Singular Points on Surfaces Immersed in R4 and their Generic Bifurcations
摘要：
这里描述的axiumbilic点，出现在一般的一个参数家庭的曲面沉浸在R4。在这些点的椭圆曲率的浸入，小，加西亚-索托马约尔具有相等的轴.本文评述了轴曲率线的基本性质和相应的轴点，即主、平均轴线场的奇点、轴交点和定义它们的四次微分方程。四次微分方程的Lie-Cartan向量场悬置，给出一个与Lie-Cartan曲面相切的线场（在源浸入曲面的投影丛中，该投影丛四重覆盖轴心点的穿孔邻域），其积分曲线规则地投影在轴曲率线上。在适当的Monge图中，一般公理点的位形记为E3、E4和E5，是通过研究Lie-Cartan向量场的积分曲线得到的。应用初等分歧理论研究了公理化泛点之间的转移和消元。分析了两种一般模式E^1_{34}和E^1_{45}，并根据它们在浸入空间中的一个参数的质变（分叉）解释了它们的轴向构型，着重讨论了它们与平面上向量场的鞍结分叉的密切相似性。本文的工作可以看作是对R3中曲面的Garcia-Gutierrez- Sotomayor脐分支的R4的部分推广。由于限制较少的可微性假设和独特的方法，它与Gutierrez-Guinez- Castaneda的结果有联系。
英文摘要：
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 (in the projective bundle of the source immersed surface which quadruply covers a punctured neighborhood of the axiumbilic point) whose integral curves project regularly on the lines of axial curvature. In an appropriate Monge chart the configurations of the generic axiumbilic points, denoted by E3, E4 and E5, are obtained by studying the integral curves of the Lie-Cartan vector field. Elementary bifurcation theory is applied to the study of the transition and elimination between the axiumbilic generic points. The two generic patterns E^1_{34} and E^1_{45} are analysed and their axial configurations are explained in terms of their qualitative changes (bifurcations) with one parameter in the space of immersions, focusing on their close analogy with the saddle-node bifurcation for vector fields in the plane . This work can be regarded as a partial extension to R4 of the umbilic bifurcations in Garcia - Gutierrez - Sotomayor for surfaces in R3. With less restrictive differentiability hypotheses and distinct methodology it has points of contact with the results of Gutierrez - Guinez - Castaneda.
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