A characterization of geometrical mappings of Grassmann spaces
The author continues his work in [J. Geom. 75, 132--150 (2002; Zbl 1035.51013); J. Geom. 79, No. 1--2, 169--176 (2004; Zbl 1062.51002)], and some earlier work of \textit{H. Havlicek} [Mitt. Math. Ges. Hamburg 14, 117--120 (1995; Zbl 0877.51001)] and \textit{A. Kreuzer} [Aequationes Math. 56, 243--250 (1998; Zbl 0927.51001)]. The author considers the Grassmann space \({\mathcal G}_k\), i.e. the set of \(k\)-dimensional subspaces of some projective space \(\mathcal P\) of order \(n\in {\mathbb N}\), \(0\leq k \leq n-1 \). Given a base \({\mathcal B}_0\) of \(n+1\) points of \(\mathcal P\) the base subset of \({\mathcal G}_k\) associated with \({\mathcal B}_0\) is the set of those subspaces from \({\mathcal G}_k\) which are generated by some \(S\subset {\mathcal B}_0\). An embedding is a mapping between projective spaces which preserves collinearity and non-collinearity. An embedding is called strong if, furthermore, it preserves independent sets. Let \(\mathcal P\) and \(\mathcal P'\) be projective spaces of the same dimension \(n\). The author's main result is Theorem 1: If for \(1\leq k \leq n-2\) a (not necessarily surjective) mapping \(f: {\mathcal G}_k \longrightarrow {\mathcal G}'_k\) between Grassmann spaces preserves base subsets, then for \(n\neq 2k+1\) it is induced by a strong embedding \({\mathcal P}\longrightarrow {\mathcal P'}\). If \(n=2k+1\) then \(f\) is induced by a strong embedding of \({\mathcal P}\) either into \({\mathcal P'}\) or into its dual. In both cases surjective mappings \(f\) are induced by collineations.
scientific article; zbMATH DE number 2119250