Invariant Einstein metrics on generalized Wallach spaces

Invariant Einstein metrics on generalized Wallach spaces have been classified except $SO(k+l+m)/SO(k)\times SO(l)\times SO(m)$. In this paper, we give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type $SO(k+l+m)/SO(k)\times SO(l)\times SO(m)$ admitting exactly two, three, or four Einstein invariant metrics up to a homothety.