Characterization of dual polar spaces

Abstract EN

In this paper we show that geometries satisfy the following axioms E1- E4 and (?n) with the assumption that symplecta have rank 2 are dual polar spaces.

Let * * *n , n = 2, 3,...

.Then * is a point-line geometry (P, L) that satisfies the following four axioms: (E1) * is connected ( i.e.

, the set of points P has a connected collinearity graph ) and * has thick lines.

(E2) For any positive integer k * n, every geodesic ( x0, x1, ...

, xk) of length k completes to a geodesic (x0 ,...,xk ,..., xn) of length n.

(E3) diam* := max{ d* (p, q ) * p, q * P} is exactly n and for each point p, the set *n-1 (p) :={ q * p * d* (p, q ) * n -1} is a geometric hyperplane of *.

(E4) If p and q are distinct points of * with d* (p, q ) = k then the convex closure < p, q >* is a member of *k, where (?n) for each point-symplecton pair (p, S), p ? S, **n-2(p)?S is either S or a point of S.