Ricci-flat and Einstein pseudoriemannian nilmanifolds

Abstract

This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\le 8$ are obtained. Some related open questions are presented.

arXiv comment30 pages, 1 figure. v2: added a comment on a recent example of an Einstein nilpotent Lie algebra of dimension 7; added a remark and a question concerning the characteristically nilpotent case; replaced the "σ-compatible" condition with the more general "σ-diagonal"; added 3 references

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Cited by in corpus (8)

• Construction of nice nilpotent Lie groups
• Indefinite Einstein metrics on nice Lie groups
• Diagram involutions and homogeneous Ricci-flat metrics
• Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds
• The Ricci-flatness that lurks in weight
• Uniqueness of ad-invariant metrics
• A non-Standard Indefinite Einstein Solvmanifold
• Nice pseudo-Riemannian nilsolitons