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

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

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


MSC:
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30Homogeneous manifolds (differential geometry)
22E25Nilpotent and solvable Lie groups