Polar spaces embedded in a projective space

Summary

This paper is a survey of the results related to polar spaces “embedded” into a projective space. This involves polar spaces embedded in a polarity (Veldkamp and Tits), fully embedded polar spaces (Buekenhout, Lefèvre and Dienst) and weakly embedded polar spaces (Lefèvre).

INTRODUCTION

The study of orthogonal, hermitian and symplectic quadrics led Veldkamp [14] to a notion of polar space. A simpler set of axioms was stated by Tits in order to classify buildings [13]. In 1974, Buekenhout and Shult [5] simplified Tits' axioms. They defined a polar space as an incidence structure, with non void set of points P, non void set of lines L and incidence relation I, such that, for each line L and each point p not incident with L, one of the following occurs:

(i) there exists exactly one point p' incident with L and a line L' incident to both p and p';

(ii) for each point p' incident with L, there is a line L' incident to both p and p'.

We also suppose that there is no point p of P such that each point of P is incident to a line incident with p.

A polar space (P, L, I) such that (ii) does not occur is a generalized quadrangle.

Tits [13] has classified all polar spaces which are not generalized quadrangles. (It is presently hopeless to classify the latter.) Up to a few known exceptions, they are isomorphic to the polar space associated with a pseudo-quadratic form. However, this classification does not determine all possible “representations” or “embeddings” of a polar space into a projective space.