*(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 $(\U0001d524,g)$, that is, a Lie algebra $\U0001d524$ together with an Euclidean metric $g$ which satisfy the Einstein equation.

More precisely, let $(\U0001d52b,g)$ be a metric nilpotent Lie algebra and $d$ a $g$-symmetric derivation of $\U0001d52b$. Then the solvable metric Lie algebra $(\U0001d524,\tilde{g})$, where $\U0001d524=\mathbb{R}d+\U0001d52b$ is a semidirect sum and $\tilde{g}$ is a natural extension of the metric $g$ such that $\tilde{g}(d,d)=1$, $\tilde{g}(d,\U0001d52b)=0$ is called a metric rank one solvable extension of $(\U0001d52b,g)$. It is known that if $(\U0001d524,\tilde{g})$ is an Einstein metric Lie algebra which contains $(\U0001d52b,g)$ as a codimension one (metric) nilpotent ideal, then $(\U0001d524,\tilde{g})$ is a uniquely defined metric rank one solvable extension of $(\U0001d524,g)$.

The author classifies all metric rank one extensions of 6-dimensional metric nilpotent Lie algebras $(\U0001d52b,g)$ 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 |