Affine automorphisms that are isometries (Q1870067)

The Grassmann space \(\text{G}(k,n) \subset \text{PG}_N{\mathbb R}\), \(N= {n \choose k}-1, \) consists of all \(k\)-dimensional linear subspaces of \({\mathbb R}^n\). A family \(\mathfrak L \subset \text{G}(k,n)\) is called co-conical if \(\mathfrak L\) is contained in a hyperquadric \(Q\) such that \(\text{G}(k,n) \not\subseteq Q\). Affine subspaces are co-conical, if the corresponding linear subspaces are. The author proves the following theorem: If \({\mathfrak L} \subset \text{G}(k,n)\) is not co-conical and if the affine automorphism \(F: {\mathbb R}^n \to {\mathbb R}^n\) preserves the \(k\)-dimensional volume on each \(L \in {\mathfrak L}\), then \(F\) is an isometry.