Affine polar spaces are polar spaces from which a hyperplane (that is a proper subspace meeting every line of the space) has been removed. These spaces are of interest as they constitute quite natural examples of 'locally polar spaces'. A characterization of affine polar spaces (of rank at least 3) is given as locally polar spaces whose planes are affine. Moreover, the affine polar spaces are fully classified in the sense that all hyperplanes of the fully classified polar spaces (of rank at least 3) are determined.
O. INTRODUCTIONIn 1959, Veldkamp [9] initiated the synthetic study of geometries induced on the set of absolute points, lines, planes, etc. with respect to a polarity, and named the subject polar geometry. After subsequent work of Tits [7], Buekenhout and Shult [2] and Buekenhout and Sprague [3] a somewhat larger class of point, line geometries emerged which could be characterized by the beautiful axiom If p is a point and L a line, then the set of points incident with L and collinear with p is either a singleton or the set of all points incident with L, which we shall quote as the 'one or all' axiom. An incidence system (P, ~) [-i.e. a pair consisting of a set P (of points) and set ~q~ (of lines) together with a relation between them, called incidence, such that each line is incident with at least two points] is called a polar space if the 'one or all' axiom is satisfied. An incidence system is called nondegenerate if no point is collinear with all others, and it is called singular if any two of its points are collinear. If X is a subset of the point set P of the incidence system (P, ~) and L ~ ~q~, we denote by X(L) the set of points in X incident to L, and by ~q~(X) the set of all lines in incident to at least two points of X. Thus, 5¢(X) --{LeSI IX(L)] > 1}.Restricting incidence of (P, ~#), we can regard (X, 5°(X)) as an incidence system. If each point incident to a line in ~(X) belongs to X, we say that X is a subspace of (P, 5(). A subspace of a polar space is again a polar space. The singular rank of an incidence system (P, ~o) is the maximal number n (possibly oo) for which there exists a chain of distinct subspaces *E.E.S. was partially supported by the National Science Foundation, U.S.A. 35:43 76, 1990.
Geometriae Dedicata