Metric characterization of apartments in dual polar spaces

Let Pi be a polar space of rank n and let G(k)(Pi), k is an element of {0, ..., n - 1} be the polar Grassmannian formed by k-dimensional singular subspaces of Pi. The corresponding Grassmann graph will be denoted by Gamma(k)(Pi). We consider the polar Grassmannian G(n - 1)(Pi) formed by maximal singular subspaces of Pi and show that the image of every isometric embedding of the n-dimensional hypercube graph H-n in Gamma(n - 1)(Pi) is an apartment of G(n - 1)(Pi). This follows from a more general result concerning isometric embeddings of H-m, m <= n in Gamma(n - 1)(Pi). As an application, we classify all isometric embeddings of Gamma(n - 1)(Pi) in Gamma(n' - 1)(Pi'), where Pi' is a polar space of rank n' >= n. (C) 2010 Elsevier Inc. All rights reserved.