Homogeneous Einstein metrics

We consider homogeneous Einstein metrics on symmetric spaces and we describe their geometry. For compact irreducible symmetric spaces with rank($M) \u3e 1$, not isometric to a Lie group, we classify all homogeneous Einstein metrics. Whenever there exists a closed proper subgroup $G$ of Isom($M$) acting transitively on $M$ we find all $G$-homogeneous (non-symmetric) Einstein metrics on $M$. We next examine three reducible symmetric spaces on which a simple Lie group $G$ acts transitively, and we describe all $G$-invariant Einstein metrics. Finally, we discuss the SO($n + 1$)-invariant Einstein metrics on the Stiefel manifold SO($n + 1$)/SO($n - 1) \cong V\sb2(\IR\sp{n + 1}$)