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Ioannis Chrysikos

Publications and source records attributed to Ioannis Chrysikos.

30 records · Page 2Linked to original sources
arXiv2015

Non-naturally reductive Einstein metrics on exceptional Lie groups

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Gröbner bases and classify the real solutions of the associated algebraic systems. For the Lie group ${\rm G}_2$ we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups ${\rm E}_7$ and ${\rm E}_8$, we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are ${\rm Ad}(K)$-invariant by different Lie subgroups $K$.

math.DG↗
arXiv2015

Killing and twistor spinors with torsion

We study twistor spinors (with torsion) on Riemannian spin manifolds $(M^{n}, g, T)$ carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection $\nabla^{c}=\nabla^{g}+\frac{1}{2}T$ and under the condition $\nabla^{c}T=0$, we show that the twistor equation with torsion w.r.t. the family $\nabla^{s}=\nabla^{g}+2sT$ can be viewed as a parallelism condition under a suitable connection on the bundle $Σ\oplusΣ$, where $Σ$ is the associated spinor bundle. Consequently, we prove that a twistor spinor with torsion has isolated zero points. Next we study a special class of twistor spinors with torsion, namely these which are $T$-eigenspinors and parallel under the characteristic connection; we show that the existence of such a spinor for some $s\neq 1/4$ implies that $(M^{n}, g, T)$ is both Einstein and $\nabla^{c}$-Einstein, in particular the equation ${\rm Ric}^{s}=\frac{{\rm Scal}^{s}}{n}g$ holds for any $s\in\mathbb{R}$. In fact, for $\nabla^{c}$-parallel spinors we provide a correspondence between the Killing spinor equation with torsion and the Riemannian Killing spinor equation. This allows us to describe 1-parameter families of non-trivial Killing spinors with torsion on nearly Kähler manifolds and nearly parallel ${\rm G}_2$-manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere ${\rm S}^{3}$. We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.

math.DG↗
arXiv2015

Invariant connections with skew-torsion and $\nabla$-Einstein manifolds

For a compact connected Lie group $G$ we study the class of bi-invariant affine connections whose geodesics through $e\in G$ are the 1-parameter subgroups. We show that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra $\frak{g}$ coincide with the bi-invariant metric connections. Next we describe the geometry of a naturally reductive space $(M=G/K, g)$ endowed with a family of $G$-invariant connections $\nabla^α$ whose torsion is a multiple of the torsion of the canonical connection $\nabla^{c}$. For the spheres ${\rm S}^{6}$ and ${\rm S}^{7}$ we prove that the space of ${\rm G}_2$ (resp. ${\rm Spin}(7)$)-invariant affine or metric connections consists of the family $\nabla^α$. Then we examine the "constancy" of the induced Ricci tensor ${\rm Ric}^α$ and prove that any compact simply-connected isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a $\nabla^α$-Einstein manifold for any $α\in\mathbb{R}$. We also provide examples of $\nabla^{\pm 1}$-Einstein structures for a class of compact homogeneous spaces $M=G/K$ with two isotropy summands.

math.DG↗
arXiv2012

Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands

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 new technique which is based on a fibration of a flag manifold over another flag manifold and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E_6/(SU(4) x SU(2) x U (1) x U (1)) and E_7/(U(1) x U(6)) we find explicitely all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2\ell +1)/(U(1) x U (p) x SO(2(\ell -p-1)+1)) and SO(2\ell)/(U(1) x U (p) x SO(2(\ell -p-1))) we prove existence of at least two non Kähler-Einstein metrics. For small values of $\ell$ and $p$ we give the precise number of invariant Einstein metrics.

math.DG↗
arXiv2012

The classification of homogeneous Einstein metrics on flag manifolds with $b_2(M)=1$

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 $G/H$ corresponding to a classical simple Lie group $G$ and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler--Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds $\E_8/\U(1)\times \SU(4)\times \SU(5)$ and $\E_8/\U(1)\times \SU(2)\times \SU(3)\times \SU(5)$. In order to determine explicitly the Ricci tensors of an $\E_8$-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold $G/H$ with $b_{2}(M)=1$ admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

math.DG↗
arXiv2012

Flag manifolds, symmetric $\fr{t}$-triples and Einstein metrics

Let $G$ be a compact connected simple Lie group and let $M=G^{\bb{C}}/P=G/K$ be a generalized flag manifold. In this article we focus on an important invariant of $G/K$, the so called $\fr{t}$-root system $R_{\fr{t}}$, and we introduce the notion of symmetric $\fr{t}$-triples, that is triples of $\fr{t}$-roots $ξ, ζ, η\in R_{\fr{t}}$ such that $ξ+η+ζ=0$. We describe their properties and we present an interesting application on the structure constants of $G/K$, quantities which are straightforward related to the construction of the homogeneous Einstein metric on $G/K$. Next we classify symmetric $\fr{t}$-triples for generalized flag manifolds $G/K$ with second Betti number $b_{2}(G/K)=1$, and we treat also the case of full flag manifolds $G/T$, where $T$ is a maximal torus of $G$. In the last section we construct the homogeneous Einstein equation on flag manifolds $G/K$ with five isotropy summands, determined by the simple Lie group $G=\SO(7)$. By solving the corresponding algebraic system we classify all $\SO(7)$-invariant (non-isometric) Einstein metrics, and these are the very first results towards the classification of homogeneous Einstein metrics on flag manifolds with five isotropy summands.

math.DG↗
arXiv2010

Homogeneous Einstein metrics on G_2/T

We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus $G_2/T$ turns out to be the first known example of an exceptional full flag manifold which admits at least one non-Kähler and not normal homogeneous Einstein metric.

math.DG↗
arXiv2010

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

We give the global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces descents to a parameter depending system of two or three ordinary differential equations, respectively. We present here the qualitative study of these system's global phase portrait, by using techniques of Dynamical Systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds, and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.

math.DG↗
arXiv2009

Invariant Einstein metrics on flag manifolds with four isotropy summands

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 geometry. We present new $G$-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.

math.DG↗
arXiv2009

Invariant Einstein metrics on generalized flag manifolds with two isotropy summands

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 determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

math.DG↗