论文标题
r^4中表面的反射图和几何形状
Reflexion maps and geometry of surfaces in R^4
论文作者
论文摘要
在本文中，我们介绍了新的不变积分---“特殊抛物线点” ---在真实4空间中的抛物线套寄生er集合集中，与2-参数的反射家族中的对称性相关。以这种方式检测到抛物线套件本身，并给出每个弧线的符号，在特殊点发生变化，其中家庭具有额外的对称程度。反射家族检测到的$ m $的其他点包括真实和虚构类型的屈光点，其中首先也与抛物线寄生虫集的标志更改有关。我们展示了如何以Monge形式给出$ M $的情况，并举例说明了一些示例，说明了在1参数表面家族中的特殊寄托点的诞生。我们从奇异理论中使用的工具是从平面到平面的某些对称地图的接触分类，我们给出了该分类的开始，包括我们与$ M $的几何形状相关的广泛展开。
In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given a sign, which changes at the special points, where the family has an additional degree of symmetry. Other points of $M$ which are detected by the family of reflexions include inflexion points of real and imaginary type, and the first of these is also associated with sign changes on the parabolic set. We show how to compute the special points globally for the case where $M$ is given in Monge form and give some examples illustrating the birth of special parbolic points in a 1-parameter family of surfaces. The tool we use from singularity theory is the contact classification of certain symmetric maps from the plane to the plane and we give the beginning of this classification, including versal unfoldings which we relate to the geometry of $M$.
