Metric characterization of apartments in dual polar spaces

Let $Π$ be a polar space of rank $n$ and let ${\mathcal G}_{k}(Π)$, $k\in \{0,\dots,n-1\}$ be the polar Grassmannian formed by $k$-dimensional singular subspaces of $Π$. The corresponding Grassmann graph will be denoted by $Γ_{k}(Π)$. We consider the polar Grassmannian ${\mathcal G}_{n-1}(Π)$ formed by maximal singular subspaces of $Π$ and show that the image of every isometric embedding of the $n$-dimensional hypercube graph $H_{n}$ in $Γ_{n-1}(Π)$ is an apartment of ${\mathcal G}_{n-1}(Π)$. This follows from a more general result (Theorem 2) concerning isometric embeddings of $H_{m}$, $m\le n$ in $Γ_{n-1}(Π)$. As an application, we classify all isometric embeddings of $Γ_{n-1}(Π)$ in $Γ_{n'-1}(Π')$, where $Π'$ is a polar space of rank $n'\ge n$ (Theorem 3).

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