Surjections on Grassmannians preserving pairs of elements with bounded distance

Let m and k be two fixed positive integers such that m>k⩾2. Let V be a left vector space over a division ring with dimension at least m+k+1. Let Gm(V) be the Grassmannian consisting of all m-dimensional subspaces of V. We characterize surjective mappings T from Gm(V) onto itself such that for any A,B in Gm(V), the distance between A and B is not greater than k if and only if the distance between T(A) and T(B) is not greater than k.