Isometric embeddings of Johnson graphs in Grassmann graphs

Let V be an n-dimensional vector space (4 ≤ n < ∞) and let Gk(V ) be the Grassmannian formed by all k-dimensional subspaces of V . The corresponding Grassmann graph will be denoted by Γk(V ). We describe all isometric embeddings of Johnson graphs J (l,m), 1 < m < l − 1 in Γk(V ), 1 < k < n − 1 (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of J (n, k) in Γk(V ) is an apartment of Gk(V ) if and only if n = 2k. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in Γk(V ), 1 < k < n− 1.