# Polar space

Let $P$ be a set of points with a non-empty collection of distinguished subsets of cardinality $\geq 2$, called lines. Such a structure is called a polar space if for each line $l$ of $P$ and each point $A \in P \setminus l$ the point $A$ is collinear either with precisely one or with all points of $l$. A non-degenerate polar space is one which has no points that are collinear with other points (i.e. it is not a "cone" ). A polar space is linear if two distinct lines have at most one common point.

Examples arise by taking a projective space $\mathbf P ^ {d}$( $d \geq 3$ to get something non-trivial) with a polarity defined by a non-degenerate bilinear form $Q$. Take the subset $P$ of absolute points (also called isotropic points), i.e. $P = \{ {x \in \mathbf P ^ {d} } : {Q ( x , x ) = 0 } \}$. The lines in $P$ are the projective lines of $\mathbf P ^ {d}$ which are entirely in $P$. The name "polar space" derives from this class of examples.

A subspace of a polar space is a subset $P ^ \prime$ of $P$ such that if $A, B \in P ^ \prime$ and $A$ and $B$ are collinear and unequal, then the whole line through $A$ and $B$ is in $P ^ \prime$. A singular subspace of a polar space is one in which every pair of points of it is collinear.

A Tits polar space of rank $n$, $n \geq 2$, is a set of points $P$ together with a family of subsets, called subspaces, such that:

i) a subspace together with the subspaces contained in it is a $d$- dimensional projective space;

ii) the intersection of two subspaces is a subspace;

iii) given a subspace $V$ of dimension $n- 1$ and a point $A \in P \setminus V$, there is a unique subspace $W$ containing $A$ such that $V \cap W$ has dimension $n - 2$; the space $W$ contains all points of $V$ that are joined to $A$ by a line (a subspace of dimension 1);

iv) there exist at least two disjoint subspaces of dimension $n- 1$.

The Tits polar spaces of rank $\geq 3$ are known , [a2] and are classical, i.e. they are Tits polar spaces arising from a $( \sigma - \epsilon )$- Hermitian form (cf. Sesquilinear form) or a pseudo-quadratic form on a vector space over a division ring, by taking as subspaces the totally-isotropic subspaces of the form (of Witt index $\geq 2$). In particular, the subspaces of a finite polar space of rank $\geq 3$ are the totally-isotropic subspaces with respect to a polarity of a finite projective space or the projective spaces in a non-singular quadric in a finite projective space.

Every non-degenerate polar space is linear, and if for a non-degenerate polar space of finite rank all lines have cardinality $\geq 3$, then the singular subspaces define a classical polar space [a3].

A non-degenerate polar space is either classical or a generalized quadrangle (cf. Quadrangle, complete).

#### References

 [a1a] F.D. Veldkamp, "Polar geometry" Indag. Math. , 21 (1959) pp. 512–551 [a1b] F.D. Veldkamp, "Polar geometry" Indag. Math. , 22 (1960) pp. 207–212 [a2] J. Tits, "Buildings and BN-pairs of spherical type" , Springer (1974) pp. Chapt. 8 [a3] F. Buekenhout, E.E. Shult, "On the foundations of polar geometry" Geom. Dedicata , 3 (1974) pp. 155–170 [a4] R. Dembowski, "Finite geometries" , Springer (1968) pp. 254
How to Cite This Entry:
Polar space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_space&oldid=48229