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Einstein solvmanifolds and nilsolitons. (English) Zbl 1186.53058
Gordon, Carolyn S. (ed.) et al., New developments in Lie theory and geometry. Proceedings of the 6th workshop on Lie theory and geometry, Cruz Chica, Córdoba, Argentina, November 13–17, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4651-3/pbk). Contemporary Mathematics 491, 1-35 (2009).

The only known examples of noncompact homogeneous Riemannian manifolds which admit invariant Einstein metric are simply connected solvable Lie groups with a left invariant Riemannian metric (solvmanifolds). There is a conjecture that these examples exhaust all the possibilities and the conjecture is true up to dimension 5.

The paper gives a detailed report on the present status of the study of Einstein solvmanifolds.

A generalization of Einstein metrics is the notion of Ricci soliton. If $S$ is an Einstein solvmanifolds, then the metric restricted to the submanifold $N=\left[S,S\right]$ is a Ricci soliton and conversely, any Ricci soliton left invariant metric on a nilpotent Lie group $N$ (a nilsoliton) can be extended to an Einstein solvmanifold. The problem of classification of Einstein solvmanifolds is equivalent to the classification of Ricci solitons on nilpotent Lie groups.

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