Related papers: Einstein metrics on the full flag $F(n)$

A generalized flag manifold is a homogeneous space of the form $G/K$, where $K$ is the centralizer of a torus in a compact connected semisimple Lie group $G$. We classify all flag manifolds with four isotropy summands and we study their…

It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to an algebraic system if $g$ is a $G$-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag…

We construct the homogeneous Einstein equation for generalized flag manifolds $G/K$ of a compact simple Lie group $G$ whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a…

We obtain new invariant Einstein metrics on the compact Lie group $\SU(N)$ which are not naturally reductive. This is achieved by using the generalized flag manifold $G/K=\SU(k_1+\cdots +k_p)/\s(\U(k_1)\times\cdots\times\U(k_p))$ and by…

The classification of compact homogeneous spaces of the form $M=G/K$, where $G$ is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are $4$ infinite families and $3$ isolated spaces…

We find the precise number of non-K\"ahler $SO(2n)$-invariant Einstein metrics on the generalized flag manifold $M=SO(2n)/U(p)\times U(n-p)$ with $n\geq 4$ and $2\leq p\leq n-2$. We use an analysis on parametric systems of polynomial…

We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces $\mathrm{SU}(n)$, $n\geq3$, and $E_6/F_4$. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces…

We find the precise number of non-K\"ahler $Sp(n)$-invariant Einstein metrics on the generalized flag manifold $M=Sp(n)/(U(p)\times U(n-p))$ with $n\geq 3$ and $1\leq p\leq n-1$. We use an analysis on parametric systems of polynomial…

We consider the homogeneous space $M=H\times H/\Delta K$, where $H/K$ is an irreducible symmetric space and $\Delta K$ denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of $H\times H$-invariant…

It is well known that every compact simple group manifold G admits a bi-invariant Einstein metric, invariant under G_L\times G_R. Less well known is that every compact simple group manifold except SO(3) and SU(2) admits at least one more…

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…

We study invariant Einstein metrics on the Stiefel manifold $V_k\mathbb{R}^n\cong \mathrm{SO}(n)/\mathrm{SO}(n-k)$ of all orthonormal $k$-frames in $\mathbb{R}^n$. The isotropy representation of this homogeneous space contains equivalent…

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…

We study homogeneous Einstein metrics on indecomposable non-K\"ahlerian C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank} G>\mathsf{rank} H$ over flag manifolds $F=G/K$ of a compact simple Lie group $G$. Based on the…

We study the linear stability of Einstein metrics of Riemannian submersion type. First, we derive a general instability condition for such Einstein metrics and provide some applications. Then we study instability arising from Riemannian…

In this paper we employ numerical methods to study the Einstein equation \[ Ric(g)=\lambda\, g, \] where $Ric$ is the Ricci tensor and $\lambda$ is the Einstein constant, restricted to a class of full flag manifolds. These metrics describe…

We consider invariant Einstein metrics on the Stiefel manifold $V_q\bb{R} ^n$ of all orthonormal $q$-frames in $\bb{R}^n$. This manifold is diffeomorphic to the homogeneous space $\SO(n)/\SO(n-q)$ and its isotropy representation contains…

Let $G$ be a simple compact connected Lie group. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds $G/H$ with second Betti number $b_{2}(G/H)=1$. There are 8 infinite families…

We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible…