I would ask if the geometry (in Klein's sense) of bijective applications f such that (considering as example a metrical space (M,d) and four arbitrary elements a,b,c,d of M) d(a,b)<d(c,d) if and only if d(f(a),f(b))<d(f(c),f(d)) has been ever studied. I think that such a geometry is too important to be possible that it has been never studied. GL