In mathematics, in the field of geometry, a polar space of rank n, or projective index, consists of a set P, conventionally called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms:
The intersection of two subspaces is always a subspace.
For each point p not in a subspace A of dimension of, there is a unique subspace B of dimension such that is -dimensional. The points in 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.
It is possible to define and study a slightly biggerclass of objects using only relationship between points and lines: a polar space is a partial linear space , 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 are also studied as combinatorial objects.
Generalized quadrangles
A polar space of rank two is a generalized quadrangle; in this case, in the latter definition, the set of points of a lineℓ collinear with a point p is the whole ℓ only if p ∈ ℓ. 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 ℓ and a point p not on ℓ so that p is collinear to all points of ℓ.
Let be the projective space of dimension over the finite field and let be a reflexivesesquilinear form or a quadratic form on the underlying vector space. Then the elements of the finite classical polar space associated with this form consists of the totally isotropic subspaces or the totally singular subspaces of with respect to. 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. The number of points in a is denoted by and it is equal to. When is equal to, we get a generalized quadrangle.
Form
Name
Notation
Number of points
Collineation group
Alternating
Symplectic
Hermitian
Hermitian
Hermitian
Hermitian
Quadratic
Hyperbolic
Quadratic
Parabolic
Quadratic
Elliptic
Classification
proved that a finite polar space of rank at least three, is always isomorphic with one of the three types of classical polar spaces given above. This leaves open only the problem of classifying the finite generalized quadrangles.
