It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric constant holomorphic sectional curvature. In this paper we give full classification all possible Riemannian metrics on group. We prove each those negative scalar curvature, only one them being Einstein (up to isometry and scaling).

0. Introduction. M. M. Day has shown that an abelian group has only one invariant mean if, and only if, the group is finite [3, Theorem 4, p. 534]. Knowing this we now ask to what extent the invariant means on a given abelian group are alike. In this paper we shall show that they all have the same "measure-theoretic character." An invariant mean on an abelian group G is a normalized, positive, ...