Embeddings of Grassmann graphs

Let $V$ and $V'$ be vector spaces of dimension $n$ and $n'$, respectively. Let $k\in\{2,...,n-2\}$ and $k'\in\{2,...,n'-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph $\Gamma_{k}(V)$ in the Grassmann graph $\Gamma_{k'}(V')$

Geometrical characterization of semilinear isomorphisms of vector spaces and semilinear homeomorphisms of normed spaces

Let $V$ and $V'$ be vector spaces over division rings (possible infinite-dimensional) and let ${\mathcal P}(V)$ and ${\mathcal P}(V')$ be the associated projective spaces. We say that $f:{\mathcal P}(V)\to {\mathcal P}(V')$ is a PGL-{\it mapping} if for every $h\in {\rm PGL}(V)$ there exists $h'\in {\rm PGL}(V')$ such that $fh=h'f$. We show that for every PGL-bijection the inverse mapping is a semicollineation. Also, we obtain an analogue of this result for the projective spaces associated to normed spaces