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 called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms: It is possible to define and study a slightly bigger class of objects using only the 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. Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.

A polar space of rank two is a generalized quadrangle; in this case, in the latter definition, the set of points of a line

collinear with a point p is the whole of

One recovers the former definition from the latter under the assumptions that lines have more than 2 points, points lie on more than 2 lines, and there exist a line

so that p is collinear to all points of

be the projective space of dimension

be a reflexive sesquilinear form or a quadratic form on the underlying vector space.

The elements of the finite classical polar space associated with this form are the elements of the totally isotropic subspaces (when

is a sesquilinear form) or the totally singular subspaces (when

is a quadratic form) of

The Witt index of the form is equal to the largest vector space dimension of the subspace contained in the polar space, and it is called the rank of the polar space.

These finite classical polar spaces can be summarised by the following table, where

is the dimension of the underlying projective space and

is the rank of the polar space.

Jacques Tits proved that a finite polar space of rank at least three is always isomorphic with one of the three types of classical polar space given above.

This leaves open only the problem of classifying the finite generalized quadrangles.