Related papers: Ricci-flat and Einstein pseudoriemannian nilmanifo…
We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous manifolds and discuss when this metric is complete and K\"ahler. This construction has a strong interplay…
The goal of this paper is to show that many key results found in the study of Einstein Lorentzian nilpotent Lie algebras can still hold in the more general settings of unimodular Lie algebras and (completely) solvable Lie algebras.
There are five six-dimensional nilpotent Lie groups G, which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo-Kahler, nor almost Hermitian. In this work, these Lie groups are being…
A Riemannian manifold is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We consider weakly Einstein Lie groups with a left-invariant metric which are weakly Einstein.…
We completely describe the signatures of the Ricci curvature of left-invariant Riemannian metrics on arbitrary real nilpotent Lie groups. The main idea in the proof is to exploit a link between the kernel of the Ricci endomorphism and…
Any expanding homogeneous Ricci soliton (in particular any homogeneous Einstein manifold of negative scalar curvature) can be obtained, up to isometry, from a Lie subgroup of a nilpotent Iwasawa group $N$ whose induced metric is a Ricci…
We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to…
The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily…
The aim of this paper is to classify Ricci soliton metrics on $7$-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in [Transformation Groups, Volume 17, Number 3 (2012), 639--656]. To this end, we use…
The prescribed Ricci curvature problem involves finding a Riemannian metric g that satisfies the equation ric(g) = T, where T is a fixed symmetric (0, 2)-tensor field on a differential manifold M. In this paper, we introduce the concept of…
A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric…
A Riemannian Einstein solvmanifold is called standard, if the orthogonal complement to the nilradical of its Lie algebra is abelian. No examples of nonstandard solvmanifolds are known. We show that the standardness of an Einstein metric…
We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions…
We continue the systematic study of left-invariant generalised Einstein metrics on Lie groups initiated in arXiv:2206.01157. Our approach is based on a new reformulation of the corresponding algebraic system. For a fixed Lie algebra…
It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new…
We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture…
The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by J. E. D'Atri and W. Ziller in 1979. In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie…
In this paper we prove that the compact Lie group $G_2$ admits a left-invariant Einstein metric that is not geodesic orbit. In order to prove the required assertion, we develop some special tools for geodesic orbit Riemannian manifolds. It…
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…
The existence or non-existence of Einstein metrics on 4-manifolds with non-trivial fundamental group and the relation with the underlying differential structure are analyzed. For most points $(n,m)$ in a large region of the integer lattice,…