Non-compact Einstein manifolds with symmetry

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