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 Mat. Tr., 2007, Volume 10, Number 1, Pages 164–190 (Mi mt34)

On Einstein Extensions of Nilpotent Metric Lie Algebras

Yu. G. Nikonorov

Rubtsovsk Industrial Intitute, Branch of Altai State Technical University

Abstract: The main result of the article is as follows: If a nilpotent noncommutative metric Lie algebra $(\mathfrak n,Q)$ is such that the operator $\operatorname{Id}-\frac{\operatorname{trace}(\mathrm{Ric})}{\operatorname{trace}(\mathrm{Ric}^2)}\mathrm{Ric}$ is positive definite then every Einstein solvable extension of $(\mathfrak n,Q)$ is standard. We deduce several consequences of this assertion. In particular, we prove that all Einstein solvmanifolds of dimension at most 7 are standard.

Key words: Einstein metrics, Riemannian manifolds, nilpotent metric Lie algebras, solvmanifolds.

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English version:
Siberian Advances in Mathematics, 2007, 17:3, 153–170

Bibliographic databases:

UDC: 514.765

Citation: Yu. G. Nikonorov, “On Einstein Extensions of Nilpotent Metric Lie Algebras”, Mat. Tr., 10:1 (2007), 164–190; Siberian Adv. Math., 17:3 (2007), 153–170

Citation in format AMSBIB
\Bibitem{Nik07} \by Yu.~G.~Nikonorov \paper On Einstein Extensions of Nilpotent Metric Lie Algebras \jour Mat. Tr. \yr 2007 \vol 10 \issue 1 \pages 164--190 \mathnet{http://mi.mathnet.ru/mt34} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2485371} \elib{http://elibrary.ru/item.asp?id=9483459} \transl \jour Siberian Adv. Math. \yr 2007 \vol 17 \issue 3 \pages 153--170 \crossref{https://doi.org/10.3103/S1055134407030017} 

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This publication is cited in the following articles:
1. A. G. Kremlyov, Yu. G. Nikonorov, “The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The nonunimodular case”, Siberian Adv. Math., 20:1 (2010), 1–57
2. Nikolayevsky Y., “Einstein solvmanifolds and the pre-Einstein derivation”, Trans. Amer. Math. Soc., 363:8 (2011), 3935–3958
3. Nikonorov Yu.G., “Negative Eigenvalues of the Ricci Operator of Solvable Metric Lie Algebras”, Geod. Dedic., 170:1 (2014), 119–133
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