The existence of invariant Einstein metrics on a compact homogeneous space

Abstract:

It is proved that if the triangularizable compact set $C=X_{G,H}^{\Sigma }$ introduced by Böhm (Böhm’s polyhedron) is non-contractible, then there exists a positive-definite invariant Einstein metric $m$ of positive scalar curvature on a connected homogeneous space $G/H$ of a compact Lie group $G$. There is a natural continuous map of $C$ onto the flag complex $K_{B}$ of a finite graph $B$. For $C=K_B$ this gives one of the criteria proved by Böhm. Another consequence — that $m$ exists for disconnected $B$ — is a version of the Böhm–Wang–Ziller graph theorem (but now the graph may be disconnected for $\mathfrak {z(g)}\ne 0$). Furthermore, the preparatory Böhm theorems on retractions are revised, and in this connection new constructions of certain topological spaces are proposed.