Homogeneous Einstein metrics on generalized flag manifolds

A Riemannian manifold (M, g) is called Einstein, if there is some # ? R such that Ricg = #g, where Ricg is the Ricci tensor. It is well known that if (M = G/K, g) is a compact homogeneous Riemannian manifold, then the G-invariant Einstein metrics of unit volume, are the critical points of the scalar curvature function S :MG > R restricted to the space MG1 of all G-invariant metrics with volume 1. For a G-invariant Riemannian metric the Einstein equation reduces to a system of algebraic equations. The positive real solutions of this system are the G-invariant Einstein metrics on M. An important family of compact homogeneous spaces consists of the generalized flag manifolds. These are adjoint orbits of a compact semisimple Lie group. Flag manifolds of a compact connected semisimple Lie group exhaust all compact and simply connected homogeneous K¨ahler manifolds and are of the form G/C(S), where C(S) is the centralizer (in G) of a torus S in G. Such homogeneous spaces admit a finite number of G-invariant complex structures, and for any such complex structure there is a unique compatible Ginvariant K¨ahler-Einstein metric. In this thesis we classify all flag manifolds M = G/K of a compact simple Lie group G, whose isotropy representation decomposes into 2 or 4, isotropy summands. For these spaces we solve the (homogeneous) Einstein equation, and we obtain the explicit form of new G-invariant Einstein metrics. For most cases we give the classification of homogeneous Einstein metrics. We also examine the isometric problem. For the construction of the Einstein equation on certain flag manifolds with four isotropy summands, we apply for first time the twistor fibration of a flag manifold over an isotropy irreducible symmetric space of compact type. This method is new and it can be used also for other flag manifolds. For flag manifolds with two isotropy summands, we use the restricted Hessian and we characterize the new Einstein metrics as local minimum points of the scalar curvature function restricted to the space MG1 . We mention that the classification of flag manifolds with two isotropy summands gives us new examples of homogeneous spaces, for which the motion of a charged particle under the electromagnetic field, and the geodesics curves, are completely determined