Einstein metrics with nonpositive sectional curvature on extensions of Lie algebras of Heisenberg type
The main result of this interesting paper is a complete classification of simply connected Lie groups with left-invariant Einstein metrics such that the scalar curvature is strictly negative and the sectional curvature is non-positive, when the solvable Lie algebra \({\mathfrak s}\) of \(S\) is a metric Lie algebra \(\mathbb{R} A\oplus {\mathfrak n}\) of Iwasawa type with \(|A |=1\) and with nilpotent part \({\mathfrak n} ={\mathfrak n}_1 \oplus \cdots \oplus {\mathfrak n}_r\) where each \({\mathfrak n}_i\) is a generalized Heisenberg algebra and \([{\mathfrak n}_i, {\mathfrak n}_j] =\{0\}\) for \(i\neq j\). Moreover, the ones having strictly negative sectional curvature are explicitly determined. This result is inspired by and completes earlier work of \textit{T. H. Wolter} [Math. Z. 206, 457-471 (1991; Zbl 0716.53047)]. The so-called non-symmetric Damek-Ricci spaces which provide the only known examples of non-symmetric harmonic Riemannian manifolds form a special class of the above mentioned Lie groups. It was stated by J. Boggino that the Damek-Ricci spaces are symmetric if and only if the sectional curvature is strictly negative but his proof contained a gap. In this paper, the author provides a correct proof of this statement. A completely independent proof of the same result has been given recently by \textit{I. Dotti Miatello} in [Proc. Am. Math. Soc. 125, 573-578 (1997; Zbl 0873.53030)]. We refer to the lecture notes ``Generalized Heisenberg groups and Damek-Ricci harmonic spaces'' [Lecture Notes in Math. 1598 (Springer-Verlag, Berlin) (1995; Zbl 0818.53067)] by \textit{J. Berndt}, \textit{F. Tricerri} and \textit{L. Vanhecke} for more details and further references concerning the geometry of these remarkable classes of Lie groups.
scientific article; zbMATH DE number 1038322