Classical Polar Spaces

Abstract

In this chapter, we conclude the study of polar spaces. In Chap. 7 they made their appearance as line spaces connected with diagram geometries of type B n . In Chap. 8 they were shown to embed in projective spaces under some mild conditions, like the rank n being at least three and every line being on at least three maximal singular subspaces. Grassmannians of lines of a thick projective space over a non-commutative division ring are examples of nondegenerate polar spaces of rank three that do not satisfy these conditions. In Chap. 9, the polar spaces of rank at least two embedded in a projective space were shown to be subspaces of absolutes of quasi-polarities of the ambient projective space. In this chapter, we completely determine these polar spaces. The main result is Theorem 10.3.13 and Sect. 10.3 is devoted to its proof. Proposition 10.3.11 points out which nondegenerate polar spaces amongst those embedded in absolutes of quasi-polarities on projective spaces are proper subspaces of the absolutes. The new examples are generalizations of quadrics, called pseudo-quadrics, which are introduced in Sect. 10.2. They are characterized as the minimal polar spaces embeddable in an absolute that are invariant under perspectivities. The same property was exploited successfully in the proof of Theorem 9.5.7 (via Proposition 9.5.5), where the ambient projective space is 3-dimensional. Table 10.1 of Remark 10.3.15 surveys the relations between polar spaces (of finite rank and embeddable in projective spaces) and polar geometries, similarly to Table 6.1 for the projective case.