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 G(m) (V) be the Grassmannian consisting of all m-dimensional subspaces of V. We characterize surjective mappings T from g, (V) onto itself s...