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Rank-one Einstein solvmanifolds of dimension 7. (English) Zbl 1045.53032
The author classifies left invariant Einstein metrics on some class of 7-dimensional solvable Lie groups using the variational approach, proposed by J. Lauret. The problem reduces to the classification of some solvable 7-dimensional Einstein metric Lie algebras $(\frak g, g)$, that is, a Lie algebra $\frak g$ together with an Euclidean metric $g$ which satisfy the Einstein equation. More precisely, let $(\frak n ,g)$ be a metric nilpotent Lie algebra and $d$ a $g$-symmetric derivation of $\frak n$. Then the solvable metric Lie algebra $(\frak g , \widetilde g )$, where $\frak g = \Bbb R d + \frak n$ is a semidirect sum and $\widetilde g$ is a natural extension of the metric $g$ such that $\widetilde g(d,d) =1$, $\widetilde g(d,\frak n) =0$ is called a metric rank one solvable extension of $(\frak n, g)$. It is known that if $(\frak g , \tilde g)$ is an Einstein metric Lie algebra which contains $(\frak n, g)$ as a codimension one (metric) nilpotent ideal, then $(\frak g, \widetilde g)$ is a uniquely defined metric rank one solvable extension of $(\frak g, g)$. The author classifies all metric rank one extensions of 6-dimensional metric nilpotent Lie algebras $(\frak n, g)$ which are Einstein metric Lie algebras. She finds 34 metric nilpotent 6-dimensional Lie algebras, which have such extension.

53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30Homogeneous manifolds (differential geometry)
22E25Nilpotent and solvable Lie groups
Full Text: DOI
[1] Alekseevskii, D. V.: Conjugacy of polar factorizations of Lie groups. Math. sb. 84, 14-26 (1971) · Zbl 0239.22013
[2] Besse, A.: Einstein manifolds. Ergeb. math. 10 (1987) · Zbl 0613.53001
[3] Cortes, V.: Alekseevskian spaces. Differential geom. Appl. 6, 129-168 (1996)
[4] D’atri, J.: The curvature of homogeneous Siegel domains. J. differential geom. 15, 61-70 (1980)
[5] Damek, E.; Ricci, F.: Harmonic analysis on solvable extension of H-type groups. J. geom. Anal. 2, 213-248 (1992) · Zbl 0788.43008
[6] Heber, J.: Noncompact homogeneous Einstein spaces. Invent. math. 133, 279-352 (1998) · Zbl 0906.53032
[7] Helgason, S.: Differential geometry, Lie groups and symmetric spaces. (1978) · Zbl 0451.53038
[8] Lauret, J.: Ricci soliton homogeneous nilmanifolds. Math. ann. 319, 715-733 (2001) · Zbl 0987.53019
[9] Lauret, J.: Standard Einstein solvmanifolds as critical points. Quart. J. Math. 52, 463-470 (2001) · Zbl 1015.53025
[10] Lauret, J.: Finding Einstein solvmanifolds by a variational method. Math. Z. 241, 83-99 (2003) · Zbl 1015.53028
[11] Piatetskii-Shapiro, I. I.: Automorphic functions and the geometry of classical domains. (1969)
[12] Salamon, S.: Complex structures on nilpotent Lie algebras. J. pure appl. Algebra 157, 311-333 (2001) · Zbl 1020.17006