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 $\nabla^{\alpha}$ 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 $\nabla^{\alpha}$. Then we examine the "constancy" of the induced Ricci tensor ${\rm Ric}^{\alpha}$ and prove that any compact simply-connected isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a $\nabla^{\alpha}$-Einstein manifold for any $\alpha\in\mathbb{R}$. We also provide examples of $\nabla^{\pm 1}$-Einstein structures for a class of compact homogeneous spaces $M=G/K$ with two isotropy summands.Comment: 25 pages, to appear in Journal of Lie Theory. The presentation of the paper has been improved, some misprints and errors were corrected. The material regarding the flat case has been removed and an error in Theorem 4.7 has been correcte

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 $\Sigma\oplus\Sigma$, where $\Sigma$ 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 K\"ahler 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.Comment: to appear in Annals of Global Analysis and Geometry, the number of pages has been reduced to 28, minor change

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 \xi, \zeta, \eta\in R_{\fr{t}} such that $\xi+\eta+\zeta=0$. 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.Comment: 18 pages (the text has been reduced to 18 pages, some misprints has been corrected, unchanged results

A new 12\frac{1}{2}-Ricci type formula on the spinor bundle and applications

Consider a Riemannian spin manifold $(M^{n}, g)$ $(n\geq 3)$ endowed with a non-trivial 3-form $T\in\Lambda^{3}T^{*}M$, such that $\nabla^{c}T=0$, where $\nabla^{c}:=\nabla^{g}+\frac{1}{2}T$ is the metric connection with skew-torsion $T$. In this note we introduce a generalized $\frac{1}{2}$-Ricci type formula for the spinorial action of the Ricci endomorphism ${\rm Ric}^{s}(X)$, induced by the one-parameter family of metric connections $\nabla^{s}:=\nabla^{g}+2sT$. This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schr\"odinger-Lichnerowicz formula related to the square of the Dirac operator $D^{s}$, induced by $\nabla^{s}$, under the condition $\nabla^{c}T=0$. In the same case, we provide integrability conditions for $\nabla^{s}$-parallel spinors, $\nabla^{c}$-parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly K\"ahler manifolds and nearly parallel ${\rm G}_2$-manifolds, in dimensions 5, 6 and 7, respectively

The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1b_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 K\"ahler--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 K\"ahler). 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-K\"ahler invariant Einstein metrics. The precise number of these metrics is given in Table 1.Comment: 17 pages, (the complete classification of homogeneous Einstein metrics on flag manifolds with second Betti number 1 has been obtained

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 Gr\"obner 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 $K$.Comment: 33 page

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

A note on the volume of ∇\nabla-Einstein manifolds with skew-torsion

We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville, related with the first variation of the volume on a compact Einstein manifold.Comment: 7 pages, to appear in "Communications in Mathematics

Homogeneous Einstein metrics on the generalized flag manifold Sp(n)/(U(p) x U(n-p))

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 equations and we give some insight towards the study of such systems.Comment: 15 page

Spin structures on compact homogeneous pseudo-Riemannian manifolds

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin