Polar space

In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a set P, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms:

Every subspace is isomorphic to a projective geometry R^d(K) with −1 ≤ d ≤ (n − 1) and K a division ring. By definition, for each subspace the corresponding d is its dimension.
The intersection of two subspaces is always a subspace.
For each point p not in a subspace A of dimension of n − 1, there is a unique subspace B of dimension n − 1 such that A ∩ B is (n − 2)-dimensional. The points in A ∩ B are exactly the points of A that are in a common subspace of dimension 1 with p.
There are at least two disjoint subspaces of dimension n − 1.

It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space (P,L), so that for each point p ∈ P and each line l ∈ L, the set of points of l collinear to p, is either a singleton or the whole l.

A polar space of rank two is a generalized quadrangle; in this case in the latter definition the set of points of a line l collinear to a point p is the whole l only if p ∈ l. One recovers the former definition from the latter under assumptions that lines have more than 2 points, points lie on more than 2 lines, and there exist a line l and a point p not on l so that p is collinear to all points of l.

Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.

Examples

In a finite projective space PG(d, q) over the field of size q, with d odd and d ≥ 3, the set of all points, with as subspaces the totally isotropic subspaces of an arbitrary symplectic polarity, forms a polar space of rank (d + 1)/2.
Let Q be a nonsingular quadric in PG(n, q) with character ω. Then the index of Q will be g = (n + w − 3)/2. The set of all points on the quadric, together with the subspaces on the quadric, forms a polar space of rank g + 1.
Let H be a nonsingular Hermitian variety in PG(n, q²). The index of H will be $\left\lfloor {\frac {n-1}{2}}\right\rfloor$ . The points on H, together with the subspaces on it, form a polar space of rank $\left\lfloor {\frac {n+1}{2}}\right\rfloor$ .

Classification

Jacques Tits proved that a finite polar space of rank at least three, is always isomorphic with one of the three structures given above. This leaves only the problem of classifying generalized quadrangles.

References

Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes, 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019
Buekenhout, Francois; Cohen, Arjeh M. (2013), Diagram Geometry (Related to classical groups and buildings), A Series of Modern Surveys in Mathematics, part 3, 57, Heidelberg: Springer, MR 3014979
Buekenhout, Francis, Prehistory and History of Polar Spaces and of Generalized Polygons (PDF)

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