Low-dimensional Homogeneous Einstein Manifolds

A closed Riemannian manifold (Mn, g) is called Einstein if the Ricci tensor of g is a multiple of itself; that is, ric(g) = λ · g. This equation, called the Einstein equation, is a complicated system of second order partial differential equations, and at the present time no general existence results for Einstein metrics are known. However, there are results for many interesting classes of Einstein metrics, such as Kähler-Einstein metrics [Yau], [Tia], metrics with small holonomy group [Jo], Sasakian-Einstein metrics [BoGa] and homogeneous Einstein metrics [He], [BWZ] (cf. [Bes], [LeWa] for many more details and examples). We investigate the Einstein equation for G-invariant metrics on compact homogeneous spaces. Due to this symmetry assumption the Einstein equation becomes a system of non-linear algebraic equations which in special instances can be solved explicitly, whereas in general this seems to be impossible. Yet there are some general existence results on compact homogeneous Einstein manifolds [WZ2], [BWZ], [Bö], based on the variational characterization of Einstein metrics by the Hilbert action [Hi]. These results turn out to be very helpful in proving the following: