Characterization of isometric embeddings of Grassmann graphs

Let $V$ be an $n$-dimensional left vector space over a division ring $R$. We write ${\mathcal G}_{k}(V)$ for the Grassmannian formed by $k$-dimensional subspaces of $V$ and denote by $\Gamma_{k}(V)$ the associated Grassmann graph. Let also $V'$ be an $n'$-dimensional left vector space over a division ring $R'$. Isometric embeddings of $\Gamma_{k}(V)$ in $\Gamma_{k'}(V')$ are classified in \cite{Pankov2}. A classification of $J(n,k)$-subsets in ${\mathcal G}_{k'}(V')$, i.e. the images of isometric embeddings of the Johnson graph $J(n,k)$ in $\Gamma_{k'}(V')$, is presented in \cite{Pankov1}. We characterize isometric embeddings of $\Gamma_{k}(V)$ in $\Gamma_{k'}(V')$ as mappings which transfer apartments of ${\mathcal G}_{k}(V)$ to $J(n,k)$-subsets of ${\mathcal G}_{k'}(V')$. This is a generalization of the earlier result concerning apartments preserving mappings \cite[Theorem 3.10]{Pankov-book}.