Metric characterization of apartments in dual polar spaces

Let Π be a polar space of rank n and let G k ( 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 Γ k ( Π ) . We consider the polar Grassmannian 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 G n − 1 ( Π ) . This follows from a more general result concerning isometric embeddings of H m , m ⩽ 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 ′ ⩾ n .