  For Einstein manifolds with negative scalar curvature admitting an isometric
action of a Lie group G with compact, smooth orbit space, we show the following
rigidity result: The nilradical N of G acts polarly, and 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.
