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:

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.

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.

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