Invariant Einstein metrics on generalized Wallach spaces

TL;DR

This paper surveys invariant Einstein metrics on generalized Wallach spaces and proves that infinitely many such spaces of a specific type admit a small number of Einstein metrics up to scaling.

Contribution

It provides a comprehensive survey and establishes the existence of infinitely many spaces with exactly two, three, or four Einstein metrics in a particular class.

Findings

Infinitely many spaces admit exactly two Einstein metrics.

Infinitely many spaces admit exactly three Einstein metrics.

Infinitely many spaces admit exactly four Einstein metrics.

Abstract

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.