On bijections that preserve complementarity of subspaces

The set G of all m-dimensional subspaces of a 2m-dimensional vector space V is endowed with two relations, complementarity and adjacency. We consider bijections from G onto G^′, where G^′ arises from a 2m^′-dimensional vector space V^′. If such a bijection ? and its inverse leave one of the relations from above invariant, then also the other. In case m?2 this yields that ? is induced by a semilinear bijection from V or from the dual space of V onto V^′.As far as possible, we include also the infinite-dimensional case into our considerations.