Related papers: Einstein solvmanifolds have maximal symmetry

A Riemannian manifold (M,g) is said to be Einstein if its Ricci tensor satisfies ric(g) = cg, for some real number c. In the homogeneous case, a problem that is still open is the so called Alekseevskii Conjecture. This conjecture says that…

This is a collection of notes on the properties of left-invariant metrics on the eight-dimensional compact Lie group SU(3). Among other topics we investigate the existence of invariant pseudo-Riemannian Einstein metrics on this manifold. We…

We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $\mathrm{GL}(n,\mathbb{R})$ action, whose…

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…

The classification of homogeneous compact Einstein manifolds in dimension six is an open problem. We consider the remaining open case, namely left-invariant Einstein metrics $g$ on $G = \mathrm{SU}(2) \times \mathrm{SU}(2) = S^3 \times…

This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous…

Lying at the intersection of Ado's theorem and the Nash embedding theorem, we consider the problem of finding faithful representations of Lie groups which are simultaneously isometric embeddings. Such special maps are found for a certain…

When the maximal isometry group of a four-dimensional spacetime acts simply transitively, such a Ricci-flat metric is uniquely determined to be the Petrov solution. This isometry group is almost abelian; that is, its Lie algebra contains an…

The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent…

We study locally conformal calibrated $G_2$-structures whose underlying Riemannian metric is Einstein, showing that in the compact case the scalar curvature cannot be positive. As a consequence, a compact homogeneous $7$-manifold cannot…

We describe all Lorentzian semi-direct extensions of the Heisenberg group which are conformally Einstein. As a by side result, Bach-flat left-invariant Lorentzian metrics on semi-direct extensions of the Heisenberg group are classified,…

We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are…

We show that the system of vacuum Einstein equations (i.e., Ricci-flat metrics) with two hypersurface-orthogonal, commuting Killing vector fields in $d \ge 5$ dimensions is invariant under the action of a one-parameter Lie group, and the…

An indecomposable Lie group with Riemannian bi-invariant metric is always simple and hence Einstein. For indefinite metrics this is no longer true, not even for simple Lie groups. We study the question of whether a semi-Riemannian…

An explicit one-parameter Lie point symmetry of the four-dimensional vacuum Einstein equations with two commuting hypersurface-orthogonal Killing vector fields is presented. The parameter takes values over all of the real line and the…

Let $G$ be a compact connected Lie group and $H$ a closed subgroup of $G$. Suppose the homogeneous space $G/H$ is effective and has dimension 3 or higher. Consider a $G$-invariant, symmetric, positive-semidefinite, nonzero (0,2)-tensor…

With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such…

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy…

In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and B\'{e}rard Bergery, there are two typical types of such metrics, which…

The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds,…