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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 $\left(𝔤,g\right)$, that is, a Lie algebra $𝔤$ together with an Euclidean metric $g$ which satisfy the Einstein equation.

More precisely, let $\left(𝔫,g\right)$ be a metric nilpotent Lie algebra and $d$ a $g$-symmetric derivation of $𝔫$. Then the solvable metric Lie algebra $\left(𝔤,\stackrel{˜}{g}\right)$, where $𝔤=ℝd+𝔫$ is a semidirect sum and $\stackrel{˜}{g}$ is a natural extension of the metric $g$ such that $\stackrel{˜}{g}\left(d,d\right)=1$, $\stackrel{˜}{g}\left(d,𝔫\right)=0$ is called a metric rank one solvable extension of $\left(𝔫,g\right)$. It is known that if $\left(𝔤,\stackrel{˜}{g}\right)$ is an Einstein metric Lie algebra which contains $\left(𝔫,g\right)$ as a codimension one (metric) nilpotent ideal, then $\left(𝔤,\stackrel{˜}{g}\right)$ is a uniquely defined metric rank one solvable extension of $\left(𝔤,g\right)$.

The author classifies all metric rank one extensions of 6-dimensional metric nilpotent Lie algebras $\left(𝔫,g\right)$ which are Einstein metric Lie algebras. She finds 34 metric nilpotent 6-dimensional Lie algebras, which have such extension.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Homogeneous manifolds (differential geometry) 22E25 Nilpotent and solvable Lie groups