Polar space

Let $ P $ be a set of points with a non-empty collection of distinguished subsets of cardinality $ \geq 2 $, called lines. Such a structure is called a polar space if for each line $ l $ of $ P $ and each point $ A \in P \setminus l $ the point $ A $ is collinear either with precisely one or with all points of $ l $. A non-degenerate polar space is one which has no points that are collinear with other points (i.e. it is not a "cone" ). A polar space is linear if two distinct lines have at most one common point.

Examples arise by taking a projective space $ \mathbf P ^ {d} $( $ d \geq 3 $ to get something non-trivial) with a polarity defined by a non-degenerate bilinear form $ Q $. Take the subset $ P $ of absolute points (also called isotropic points), i.e. $ P = \{ {x \in \mathbf P ^ {d} } : {Q ( x , x ) = 0 } \} $. The lines in $ P $ are the projective lines of $ \mathbf P ^ {d} $ which are entirely in $ P $. The name "polar space" derives from this class of examples.

A subspace of a polar space is a subset $ P ^ \prime $ of $ P $ such that if $ A, B \in P ^ \prime $ and $ A $ and $ B $ are collinear and unequal, then the whole line through $ A $ and $ B $ is in $ P ^ \prime $. A singular subspace of a polar space is one in which every pair of points of it is collinear.

A Tits polar space of rank $ n $, $ n \geq 2 $, is a set of points $ P $ together with a family of subsets, called subspaces, such that:

i) a subspace together with the subspaces contained in it is a $ d $- dimensional projective space;

ii) the intersection of two subspaces is a subspace;

iii) given a subspace $ V $ of dimension $ n- 1 $ and a point $ A \in P \setminus V $, there is a unique subspace $ W $ containing $ A $ such that $ V \cap W $ has dimension $ n - 2 $; the space $ W $ contains all points of $ V $ that are joined to $ A $ by a line (a subspace of dimension 1);

iv) there exist at least two disjoint subspaces of dimension $ n- 1 $.

The Tits polar spaces of rank $ \geq 3 $ are known , [a2] and are classical, i.e. they are Tits polar spaces arising from a $ ( \sigma - \epsilon ) $- Hermitian form (cf. Sesquilinear form) or a pseudo-quadratic form on a vector space over a division ring, by taking as subspaces the totally-isotropic subspaces of the form (of Witt index $ \geq 2 $). In particular, the subspaces of a finite polar space of rank $ \geq 3 $ are the totally-isotropic subspaces with respect to a polarity of a finite projective space or the projective spaces in a non-singular quadric in a finite projective space.

Every non-degenerate polar space is linear, and if for a non-degenerate polar space of finite rank all lines have cardinality $ \geq 3 $, then the singular subspaces define a classical polar space [a3].

A non-degenerate polar space is either classical or a generalized quadrangle (cf. Quadrangle, complete).

References

[a1a]	F.D. Veldkamp, "Polar geometry" Indag. Math. , 21 (1959) pp. 512–551
[a1b]	F.D. Veldkamp, "Polar geometry" Indag. Math. , 22 (1960) pp. 207–212
[a2]	J. Tits, "Buildings and BN-pairs of spherical type" , Springer (1974) pp. Chapt. 8
[a3]	F. Buekenhout, E.E. Shult, "On the foundations of polar geometry" Geom. Dedicata , 3 (1974) pp. 155–170
[a4]	R. Dembowski, "Finite geometries" , Springer (1968) pp. 254