Metric characterization of apartments in dual polar spaces

Let Π be a polar space of rank n and let Gk(Π), k∈{0,…,n−1} be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γk(Π). We consider the polar Grassmannian Gn−1(Π) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph Hn in Γn−1(Π) is an apartment of Gn−1(Π). This follows from a more general result concerning isometric embeddings of Hm, m⩽n in Γn−1(Π). As an application, we classify all isometric embeddings of Γn−1(Π) in Γn′−1(Π′), where Π′ is a polar space of rank n′⩾n.