Related papers: Invariant Einstein metrics on basic classical Lie …

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 is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational…

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…

In this paper we study the behavior of the Ricci flow at infinity for the full flag manifold $SU(3)/T$ using techniques of the qualitative theory of differential equations, in special the Poincar\'e Compactification and Lyapunov exponents.…

It is known that every compact simple Lie group admits a bi-invariant homogeneous Einstein metric. In this paper we use two ansatz to probe the existence of additional inequivalent Einstein metrics on the Lie group SU (n) for arbitrary n.…

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor…

Let $M=G/K$ be a generalized flag manifold, that is the adjoint orbit of a compact semisimple Lie group $G$. We use the variational approach to find invariant Einstein metrics for all flag manifolds with two isotropy summands. We also…

In this note we are concerned with the distribution of Einstein and non-Einstein nilradicals among all nilpotent Lie groups. A nilpotent Lie group is called an Einstein, resp. non-Einstein, nilradical if it is a nilpotent Lie group which…

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 answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian…

We introduce two constructions to obtain left-invariant Ricci-flat pseudo-Riemannian metrics on nilpotent Lie groups, one based on gradings, the other on filtrations, both depending on the combinatorics of the set of weights. As an…

Arbitrary connections on a generic Hopf algebra $H$ are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for…

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.

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…

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…

In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita…

All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons, and nilsolitons in the…

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…

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…

This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric…