New pseudo Einstein metrics on Einstein solvmanifolds
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Hui Zhang · Zaili Yan
New pseudo Einstein metrics on Einstein solvmanifolds Received: 24 March 2020 / Accepted: 16 September 2020 Abstract. A Riemannian Einstein manifold is called an Einstein solvmanifold if there exists a transitive solvable group of isometries. In this short note, we show that every Einstein solvmanifold admits at least one pseudo-Riemannian Einstein metric.
1. Introduction Let (M, g) be a pseudo-Riemannian manifold, and the metric g is called Einstein if it satisfies ricg = cg, where ricg denotes the (0, 2)-type Ricci tensor, c ∈ R is a constant. Einstein metrics have many applications to physics as well as to other scientific fields [11,28]. Hence their study has always been one of the central problems in differential geometry. In the homogeneous Riemannian case the study is relatively more fruitful and many beautiful results have been established, see for example [1,4–6,12,15,19, 20,30–32,34] and the references therein. However, in the homogeneous pseudoRiemannian case, the study is much more complicated and the results are quite different from the Riemannian case. For example, K. Nomizu [25] constructed left invariant Lorentz metrics of constant positive, negative and zero sectional curvature on some special solvable Lie groups. Moreover, there are plenty of nilpotent Lie groups admitting bi-invariant pseudo-Riemannian metrics [2,14,29]. See [7,9,10, 13,33] for more results. As the classification of pseudo Einstein metric Lie algebras seems to be unreachable by now, it is very helpful to find more examples of such spaces. This work is supported by NSFC (Nos. 11701300, 11626134) and K.C. Wong Magna Fund in Ningbo University. H. Zhang: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China. e-mail: [email protected] Z. Yan (B): School of Mathematics and Statistics, Ningbo University, Ningbo 315211, Zhejiang, People’s Republic of China. e-mail: [email protected] Mathematics Subject Classification: 53C50 · 53C30 · 53C25
https://doi.org/10.1007/s00229-020-01249-4
H. Zhang, Z. Yan
In this paper, we construct pseudo-Riemannian Einstein metrics from Riemannian Einstein solvmanifolds. Recall that an Einstein solvmanifold (M, g) is a Riemannian Einstein manifold with a transitive solvable group of isometries. According to a deep result of [17], unless M is flat, it is simply-connected and diffeomorphic to Rn . As we are interested in the case of pseudo-Riemannian Einstein metrics of non-zero scalar curvature, we always assume that (M, g) is non-flat. Thus (M, g) can be represented as a simply-connected solvable group S with a left invariant Einstein metric g. Furthermore, a result of [24] asserts that S must be non-unimodular. For simplicity, we identify (S, g) with the metric Lie algebra (s, ·, ·), where s is the Lie algebra of S and ·, · is the metric on s induced by g. Now by the works of J. Heber [15] and J. Lauret [18,20], an Einstein non-unimodular solvable metric Lie algebra (s
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