Homogeneous Einstein Metrics and Butterflies

TL;DR

This paper explores the connection between the topology of a nerve associated with a homogeneous space and the existence of invariant Einstein metrics, providing detailed descriptions and curvature estimates.

Contribution

It offers a detailed analysis of Graev's work and curvature estimates related to Einstein metrics on homogeneous spaces.

Findings

Non-contractibility of the nerve implies existence of Einstein metrics.

Detailed description of Graev's construction and its implications.

Curvature estimates supporting the existence of Einstein metrics.

Abstract

M.~M.~Graev associated in \cite{Gr} to a compact homogeneous space $G/H$ a nerve $\XGH$, whose non-contractibility implies the existence of a $G$-invariant Einstein metric on $G/H$. The nerve $\XGH$ is a compact semi-algebraic set, defined purely Lie theoretically by intermediate subgroups. In this paper we present a detailed description of the work of Graev and the curvature estimates of \cite{Bo}.