Classification of invariant Einstein metrics on certain compact homogeneous spaces

Abstract

In this paper, we study invariant Einstein metrics on certain compact homogeneous spaces with three isotropy summands. We show that, if $G/K$ is a compact isotropy irreducible space with $G$ and $K$ simple, then except for some very special cases, the coset space $G\times~G/\Delta(K)$ carries at least two invariant Einstein metrics. Furthermore, in the case that $G_{1},~G_{2}$ and $K$ are simple Lie groups, with $K\subset~G_1,~K\subset~G_2$, and $G_{1}\neq~G_{2}$, such that $G_{1}/K$ and $G_{2}/K$ are compact isotropy irreducible spaces, we give a complete classification of invariant Einstein metrics on the coset space $G_{1}\times~G_{2}/\Delta(K)$.