Related papers: Einstein solvmanifolds and nilsolitons

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the…

Any homogeneous expanding Ricci soliton is known to be isometric to a Lie subgroup of the solvable part of the Iwasawa decomposition associated with a symmetric space of non-compact type, with the metric induced as a submanifold. In this…

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space.…

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group…

We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary…

To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on…

We call a metric $m$-quasi-Einstein if $Ric_X^m$, which replaces a gradient of a smooth function $f$ by a vector field $X$ in $m$-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein…

We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.

We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein…

In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic…

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…

In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein…

A method, due to \'Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant…

This paper presents solutions to Einstein's equation -- and the numerical methods used to construct them -- that describe simple cosmological models on manifolds with compact non-orientable spatial slices. These solutions have been…

We characteristize those Einstein four manifolds which are locally symmetric spaces of noncompact type. Namely they are four manifolds which admit solutions to the (non-Abelian) Seiberg Witten equations and satisty certain characterisitc…

We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.

An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are…

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups,…

We consider homogeneous hypercomplex manifolds with a transitive action of a compact Lie group and we give a characterization of invariant HKT metrics on them. On every such hypercomplex manifold we prove the existence of an invariant…

We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gr\"obner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly…