NON-COMPACT EINSTEIN MANIFOLDS WITH SYMMETRY

摘要

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show that the nilradical N of G acts polarly and that the N-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.