Ricci curvature and Einstein metrics on aligned homogeneous spaces

TL;DR

This paper investigates the algebraic structure of aligned homogeneous spaces with many simple factors, enabling the computation of Ricci curvature and the classification of Einstein metrics on these spaces.

Contribution

It establishes a link between topological conditions and algebraic structure, facilitating the study of Einstein metrics on aligned homogeneous spaces.

Findings

Aligned spaces have a manageable algebraic structure for Ricci curvature computation.

The paper provides criteria for the existence of Einstein metrics on these spaces.

Classification results for Einstein metrics on aligned homogeneous spaces are presented.

Abstract

Let $M=G/K$ be a compact homogeneous space and assume that $G$ and $K$ have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that $b_3(M)=s-1$ if $G$ has $s$ simple factors, so called {\it aligned}, leads to a relatively manageable algebraic structure on the isotropy representation, paving the way to the computation of Ricci curvature formulas for a large class of $G$-invariant metrics. As an application, we study the existence and classification of Einstein metrics on aligned homogeneous spaces.