Noncompact homogeneous Einstein manifolds attached to graded Lie algebras

In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is two-step. New examples of Einstein solvmanifolds with three-step and four-step nilradicals are also given. This work was partially supported by Grant-in-Aid for Young Scientists (B) 14740049 and 17740039, The Ministry of Education, Culture, Sports, Science and Technology, Japan.