The paper is devoted to the study of \(n\)-dimensional Einstein manifolds \(M\) of negative scalar curvature which admit a sufficiently large isometry group \(G\).
One of the main results is the proof of a conjecture by D. V. Alekseevskii [Mat. Sb., Nov. Ser. 96(138), 93–117 (1975; Zbl 0309.53037)]:
Conjecture. Any homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to \(\mathbb{R}^n\) and is a solvmanifold, i.e., it admits a transitive solvable group of isometries.
The above claim reduces the classification of non-compact homogeneous Einstein spaces to that of nilsolitons in the sense of J. Lauret [Math. Ann. 319, No. 4, 715–733 (2001; Zbl 0987.53019)].
The authors derive several corollaries:
(i) Any homogeneous expanding Ricci soliton is isometric to a solvsoliton and diffeomorphic to \(\mathbb{R}^n\);
(ii) Any homogeneous quaternionic Kähler manifold is either a compact symmetric Wolf space or a non-compact Alekseevsky space associated to a Clifford module;
(iii) Any compact, locally homogeneous Einstein manifold with negative scalar curvature is locally symmetric.
Surprisingly, to prove the Alekseevsky conjecture, the authors relax the homogeneity assumption as follows:
Assumption. \(M\) is a connected, complete Riemannian manifold which admits an isometry group \(G\) such that the orbit space \(B = M/G\) is a compact smooth manifold.
Then they prove the following statements.
Theorem. Let \(M\) be a Riemannian manifold of negative Ricci curvature:
(i) Then any unimodular Lie group \(G\) satisfying assumption must be non-compact semisimple;
(ii) Assume, moreover, that \(M\) is an Einstein manifold and \(G\) is a non-unimodular Lie group satisfying the above assumption. Then, the induced action of the nilradical \(N\) of \(G\) on \(M\) is polar and \(M\) is foliated into pairwise locally isometric, minimal Einstein submanifolds.
Corollary. For any \(k \geq 8\), there exist infinitely many \(k\)-dimensional, pairwise non-isomorphic Lie groups \(G\), such that for any compact manifold \(B\) of dimension \(d \geq 3\), the manifold \(M=G\times B\) admits \(G\)-invariant metrics with negative Ricci curvature, but no \(G\)-invariant Einstein metric.