Einstein Metrics with Nonpositive Sectional Curvature on Extensions of Lie Algebras of Heisenberg Type

The purpose of this article is to study some simply connected Lie groups with left invariant Einstein metric, negative Einstein constant and nonpositive sectional curvature. These Lie groups are classified if their associated metric Lie algebra s is of Iwasawa type and s = An₁n₂...n_r, where all n_iare Lie algebras of Heisenberg type with [[n_i,n_j] = {0} for ij. The most important ideas of the article are based on a construction method for Einstein spaces introduced by Wolter in 1991. By this method some new examples of Einstein spaces with nonpositive curvature are constructed. In another part of the article it is shown that Damek-Ricci spaces have negative sectional curvature if and only if they are symmetric spaces.