A characteristic property of the Grassmann manifold representing the lines of a projective space.

This paper is devoted to a characterization of the Grassmannian geometry of a projective space. The following result is proved: Let (P,L) be a partial linear space which is not linear and whose lines are not maximal singular subspaces. Suppose (P,L) has a covering Σ by maximal singular subspaces such that, for each S∈Σ and each pP−S , (1) the set of points of S collinear with p is a line L , (2) every element T of Σ through p intersects S in a unique point, and (3) each point of L belongs to an element of Σ through p . Then (P,L) is isomorphic to the Grassmannian space of a projective space. As a corollary, the authors extend to infinite spaces a characterization of Grassmannian spaces considered as Γ -spaces, which has been established by P. M. Lo Re and Olanda in the finite case