Polar Varieties in Cayley-Klein Spaces

TL;DR

This paper introduces the concept of total polars in Cayley-Klein spaces, linking them to Schubert varieties and defining spheres without relying on the space's metric, while analyzing reflections and motions.

Contribution

It defines total polars and their relation to tangent structures and motions in Cayley-Klein spaces, providing a new analytical framework for these geometries.

Findings

Total polars form Schubert varieties.

Spheres are defined independently of the metric.

Reflections are characterized as space motions.

Abstract

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.