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 algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known ...

This review article has grown out of notes for the three lectures the second author presented during the XXIV-th Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed over the last 5 years while, at the same time, we summarize all the known results. We do not give a...

‎In this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional Euclidean spaces‎, ‎which are the Einstein and M"{o}bius gyrovector spaces‎. ‎We introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and explore its...

in this paper, we obtain a necessary and sufficient condition for a conformal mapping between two weyl manifolds to preserve einstein tensor. then we prove that some basic curvature tensors of $w_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. also, we obtained the relation between the scalar curvatures of the weyl manifolds r...

It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the sa...

Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...

We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature, obtained by deforming from the quaternionic hyperbolic space of real dimension 12. We give an explicit description of this family, which is made up of Einstein solvmanifolds which share the same algebraic structure (eigenvalue type) as the rank one symmetric space HH. This deformation includes a c...

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is ηEinstein manifold if and only if base manifold is the space of constant sectional c...

We investigate the structure of conformal C-spaces, a class of Riemmanian man-ifolds which naturally arises as a conformal generalisation of the Einstein condition. A basic question is when such a structure is closed, or equivalently locally conformally Cotton. In dimension 4 we obtain a full answer to this question and also investigate the incidence of the Bach condition on this class of metri...