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}.
Categories: math.CO