A step towards the Alekseevskii conjecture

Abstract

We refine existing structure results for non-compact, homogeneous, Einstein manifolds and provide a reduction in the classification problem of such spaces. Using this work, we verify the (Generalized) Alekseevskii Conjecture for a large class of homogeneous spaces.A longstanding open question in the study of Riemannian homogeneous spaces is the classification of non-compact, Einstein spaces. In the 1970s, it was conjectured by D. Alekseevskii that any (non-compact) homogeneous Einstein space of negative scalar curvature is diffeomorphic to R n . Equivalently, this conjecture can be phrased as follows:Classical Alekseevskii Conjecture: Given a homogeneous Einstein space G/K with negative scalar curvature, K must be a maximal compact subgroup of G.

The Alekseevskii conjecture in low dimensions

“…The starting point for the proof of our main results are the structural results for non-compact homogeneous Einstein spaces given in [LL14], and specially its more recent refinements proved in [JP14]. Roughly speaking, these results state that the simply-connected cover of such a space admits a very special presentation of the form G/K, where G = (G 1 A) ⋉ N is a semi-direct product of a nilpotent normal Lie subgroup N and a reductive Lie subgroup U = G 1 A, with center A and whose semisimple part G 1 = [U, U ] has no compact simple factors and contains the isotropy K. Moreover, the orbits of U and N are orthogonal at eK, the induced metric on N is a homogeneous Ricci soliton, and the induced metric on U/K satisfies an Einstein-like condition in which the action of U on N comes into play (see (1) below).…”

“…they are isometric to a solvmanifold). We mention here that there is a stronger version of the conjecture, which is obtained by replacing the conclusion "diffeomorphic to a Euclidean space" by "isometric to a simply-connected solvmanifold" (this is commonly referred to as the strong Alekseevskii conjecture in the literature, see [JP14]). Both statements turn out to be equivalent when the isometry group is linear, and in fact at the present time all known-examples of homogeneous Einstein spaces with negative scalar curvature are isometric to simply-connected solvmanifolds.…”

“…In this case, the only information that Theorem 2.1 provides is that G 1 has no compact simple factors. (g) On the other hand, if (M, g) admits a non-semisimple transitive group of isometries, it follows from [LL14,JP14] that the group G in Theorem 2.1 may be chosen to be non-unimodular. In this case, the so called "mean curvature vector" H, implicitly defined by…”

“…Therefore by [LL14] the homogeneous space G/K satisfies all the nice properties stated in Theorem 2.1, but possibly without the additional conditions proven in [JP14] for Einstein spaces (namely, G 1 might have compact simple factors, and the action of z(u) on n might not be by symmetric endomorphisms). Nevertheless, Lemma 3.5 from [JP14] still assures that by the compatibility condition (2) one has that the family θ(z(u)) ⊂ End(n) consists of normal operators, whose transposes commute with all of θ(u). Thus, one can also consider the decomposition (4) as in the proof of Theorem 2.4, and proceed in exactly the same way to conclude that z(u) ⊥ g 1 .…”

Einstein solvmanifolds have maximal symmetry

“…For example, every compact semisimple Lie group of dimension 6 or greater admits at least two such Einstein metrics, and many admit more than this. However, in the setting of homogeneous Einstein spaces with negative Ricci curvature, there is considerably more structural rigidity; see, for example, [Heb98,LL14,JP14], and there is a reasonable hope for an eventual classfication. Presently, all known examples of homogeneous Einstein spaces of negative Ricci curvature are isometric to left-invariant metrics on solvable Lie groups, and there is mounting evidence that Einstein metrics on solvable Lie groups exhaust the class of homogeneous Einstein spaces with negative Ricci curvature.…”