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)