On the geometry of the quaternionic unit disc

In the space H of quaternions, the natural invariant geometry of the open unit disc δH, diffeomorphic to the open half-space H+ via a Cayleytype transformation, has been investigated extensively. This was accomplished by constructing, in a natural geometrical manner, the quaternionic Poincaré distance on δH (and H+). The open unit disc δH also inherits the complex Kobayashi distance when viewed as the open unit ball of C2 ∼= C + Cj ∼= H. In this paper we give an original, very simple proof of the fact that there exists no isometry between the quaternionic Poincaré distance of δH and the Kobayashi distance inherited by δH as a domain of C2. This is in accordance with the well-known consequence of the clabification theorem for the non-compact, rank 1, symmetric spaces.