proceedings of the american mathematical society Volume 119, Number 2, October 1993 ON THE RIEMANNIAN GEOMETRY OF THE NILPOTENT GROUPS H(p, r) PAOLA PIU AND MICHEL GOZE (Communicated by Jonathan M. Rosenberg) Abstract. We study some aspect of the left-invariant Riemannian geometry on a class of nilpotent Lie groups H{p, r) that generalize the Heisenberg group H2p+\ . Let us prove that the groups of type H (or Kaplan's spaces) and the H{p, r) groups have same common Riemannian properties but they are not the same spaces. Introduction The Heisenberg group H2p+i with the left-invariant metric ds2 = (dxt)2 + (dz + E*2f-i dx2i) is a typical model of a homogeneous Riemannian non-Euclidean structure. The geometry of these metrics is strongly connected to contact geometry of the Pfaff equation co- dz + E x2i-idx2i = 0. In fact, let W(co) be the group of contact transformations relative to co (i.e., of the transformations preserving the codimension 1 distribution Ker(co)). Then W(co) = Jsom(ds2), where 3som(ds2) denotes the group of isometries of ds2 . It is natural to study the Riemannian structures adapted to a generalized (i.e., of higher codimension) contact geometry. Recall that in codimension 1, every contact equation is equivalent to co = dz + 5^X2,-1^X2, = 0. This is not true anymore in codimension greater than 1, where one has an infinity of models [Gi]. In [GH] Haraguchi and the second author introduced a notion of r-contact system that seems to generalize in a remarkable way that of codimension 1 contact structure. Received by the editors March 15, 1990. 1991 Mathematics Subject Classification. Primary 53C25, 53B20. The first author is a member of the GNSAGA, CNR Italy, and of the national group "Geometria delle varieta differenziabili", 40%, M.P.I., Italy. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 611 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use