A Characterization of Grassmann Spaces of Index h of a Projective Space

In 1982/83 Bichara and Tallini characterized Grassmann spaces Gr(h,P ) of a projective space P involving intersection properties of the two disjoint families of maximal subspaces of Gr(h, P). In 1984 Melone and Olanda characterized Gr(1,P ) using only one family of maximal subspaces. In this paper the generalization of the result of Melone and Olanda to general index is given. More precisely, I prove that the natural extension of the axiom of Melone and Olanda is not sufficient to characterize Gr(h, P) when h > 1, since the affine Grassmann spaceGr (1, A) of the lines of an affine space A satisfies the axioms, too. Thus, an additional axiom is given and the characterization follows.