Gauges, or equivalently, left-invariant pseudodistances on the Heisenberg group, have been used for a long time. It was, however, only in 1978 that Cygan [3] noted that one of these natural gauges actually induces a distance, i.e., a left-invariant metric space structure on the group.
Peter Greiner posed the problem of studying the notion of arc length associated to this metric; in particular, he asked whether there is a kind of infinitesimal metric giving rise to the same arc length.
Section 2 of the present paper gives an answer to this question. It is shown that there is a certain "contravariant Riemannian metric" in the sense of [lo] (roughly speaking, a Riemannian metric in which some vectors can have infinite length), studied previously by Gaveau [6, 71, which induces the arc length in question.
We will work in the greater generality of H-type groups introduced recently by Kaplan [ll 1. This class includes among others the nilpotent parts N in the Iwasawa decomposition G = KAN of semisimple Lie groups of real rank one. In this especially interesting case the "contravariant Riemannian metric" (M,) which we construct is characterized (up to a factor) by being left-invariant under N, invariant under the centralizer M of A in K, and transforming under A by a character. It is therefore the direct generalization of the standard metric on R", which is characterized by the same properties when R" is regarded as the N-part of SO(n, 1).
In general, when an H-type group is not Abelian, (M,) arises as the limit of a family of ordinary Riemannian metrics (M,) (c > 0). In Section 3 we consider geodesic arcs with respect to these metrics. For (M,) on the Heisenberg group these were studied earlier by Gaveau [6, 71 and for (M,) by Debiard [5] and Kaplan [ 121. Besides some slight generalization of the previous results our goal here is to show that the (AI,)-geodesics joining two fixed points tend uniformly to the (M,)-geodesic as c + 0. From this one gets * Partially supported by NSF. Grant MCS-79-20062.