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 by the use of t-roots. We present new G-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.

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.2000 Mathematics Subject Classification. Primary 53C25, 57M50, Secondary 37C10.

Let M = G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for 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.2010 Mathematics subject classification: primary 53C25; secondary 53C30, 22E46.

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 F = G/H, 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.2000 Mathematics Subject Classification. 53C10, 53C30, 53C50, 53D05. Keywords: spin structure, metaplectic structure, homogeneous pseudo-Riemannian manifold, flag manifold, Koszul form, C-spaceDedicated to the memory of M. Graev 1 2 DMITRI V. ALEKSEEVSKY AND IOANNIS CHRYSIKOS Our results can be read as follows. After recalling some basic material in Section 1, in Section 2 we study invariant spin structures on pseudo-Riemannian homogeneous spaces, using homogeneous fibrations. Recall that given a smooth fibre bundle π : E → B with connected fibre F , the tangent bundle T F of F is stably equivalent to i * (T E), where i : F ֒→ E is the inclusion map (cf. [Sin]). Evaluating this result at the level of characteristic classes, one can treat the existence of a spin structure on the total space E in terms of Stiefel-Whitney classes of B and F , in the spirit of the theory developed by Borel and Hirzebruch [BoHi]. We apply these considerations for fibrations induced by a tower of closed connected Lie subgroups L ⊂ H ⊂ G (Proposition 2.6) and we describe sufficient and necessary conditions for the existence of a spin structure on the associated total space (Corollary 2.7, see also [GGO]). Next we apply these results in several particular cases. For example, in Section 3 we classify spin and metaplectic structures on compact homogeneous Kähler manifolds of a compact connected semisimple Lie group G, i.e. (generalized) flag manifolds.Generalized flag manifolds are homogeneous spaces of the form G/H, where H is the centralizer of torus in G. Here, we explain how the existence and classification of invariant spin or metaplectic structures can be treated in term of representation theory (painted Dynkin diagrams) and provide a criterion in terms of the so-called Koszul numbers (Proposition 3.12). These are the integer coordinates of the invariant Chern form (which represent the first Chern class of of an invariant complex structure J of F = G/H), with respect to the fundamental weights. By applying an algorithm given in [AℓP] (slightly revised), we compute the Koszul numbers for any flag manifold corresponding to a classical Lie group and provide necessary and sufficient conditions for the existence of a spin or metaplecti...

Abstract. 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 connectionT and under the condition ∇ c T = 0, we show that the twistor equation with torsion w.r.t. the family ∇ s = ∇ g + 2sT can be viewed as a parallelism condition under a suitable connection on the bundle Σ ⊕ Σ, 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 = 1/4 implies that (M n , g, T ) is both Einstein and ∇ c -Einstein, in particular the equation Ric s = Scal s n g holds for any s ∈ R. In fact, for ∇ 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 G 2 -manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere S 3 . We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.2000 Mathematics Subject Classification. 53C25-28, 53C30, 58J60.