Cayley-Klein空间的极簇
Polar Varieties in Cayley-Klein Spaces
摘要：
本文在分析的框架下，对Cayley-Klein空间的任意子空间引入了全极函数的概念。证明了子空间的全极集是舒伯特簇。全极坐标的概念给出了一个与空间的绝对图形相切的子空间的定义。通过指定切线，可以定义相切圆锥，然后定义球面。球面的这个定义不依赖于空间的度量。证明了由两个全极子空间定义的Cayley-Klein空间的每个反射都是该空间的运动。另一方面，n维凯莱-克莱因空间中的每个运动都是点-超平面对中至多n+1个反射的乘积。
英文摘要：
In this paper, we introduce the notion of a total polar for an arbitrary subspace of a Cayley-Klein space in an analytical framework. We show that the set of all total polars of a subspace is a Schubert variety. The notion of total polar gives a definition for a subspace to be tangent to the absolute figure of the space. By specifying tangent lines, tangent cones and then spheres are defined. This definition of the sphere does not depend on the metric of the space. It is proved that every reflection of a Cayley- Klein space, defined by two subspaces which are total polar to each other, is a motion of the space. On the other hand, each motion in a Cayley-Klein space of dimension n is a product of at most n+1 reflections in point-hyperplane pairs.
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