Non-compact Einstein manifolds with symmetry

Abstract

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G \mathsf {G} with compact, smooth orbit space, we show that the nilradical N \mathsf {N} of G \mathsf {G} acts polarly and that the N \mathsf {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.