Related papers: On pseudo-Riemannian Ricci-parallel Lie groups whi…
A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in C.R. Acad. Sci. Paris s\'er. {\bf I 333} (2001) 763-768. We study…
We classify the Ricci flat Lorentzian $n$-manifolds satisfying three particular conditions, encoding and combining some crucial features of the Kerr metrics and the Robinson-Trautman optical structures. We prove that: (a) If $n>4$, there is…
In this paper, we classify three-locally-symmetric spaces for a connected, compact and simple Lie group. Furthermore, we give the classification of invariant Einstein metrics on these spaces.
Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily…
We discuss negatively curved homogeneous spaces admitting a simply transitive group of isometries, or equivalently, negatively curved left-invariant metrics on Lie groups. Negatively curved spaces have a remarkably rich and diverse…
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For…
We define the concept of a flat pseudo-Riemannian $F$-Lie algebra and construct its corresponding double extension. This algebraic structure can be interpreted as the infinitesimal analogue of a Frobenius Lie group devoid of Euler vector…
Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures…
We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the…
In this paper, we present the classification of all possible signatures of the Ricci curvature of left-invariant Riemannian metrics on 4-dimensional Lie groups and discuss some related questions.
We continue our study of the mixed Einstein-Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a…
We investigate the conditions under which pseudo-Riemannian inner products induce pseudo-Riemannian algebraic Ricci solitons on four-dimensional Lie algebras. By analyzing the algebraic Ricci soliton equation for each four-dimensional Lie…
Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is…
A. Derdzinki [D] gave examples of Riemannian metrics with harmonic curvature and non parallel Ricci tensor on some compact manifolds $(M,g]$ . We examine their existence as well as their number wich naturally depends on the geometry of the…
In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler,…
Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. It is proved that the Ricci tensor of such a soliton is the vertical component of both B-metrics multiplied by a…
It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential…
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 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…
I discuss geometry and normal forms for pseudo-Riemannian metrics with parallel spinor fields in some interesting dimensions. I also discuss the interaction of these conditions for parallel spinor fields with the condition that the Ricci…