Einstein Homogeneous Bisymmetric Fibrations

TL;DR

This paper classifies G-invariant Einstein metrics with totally geodesic fibers on certain homogeneous fibrations involving compact semisimple Lie groups, focusing on cases with isotropy irreducibility and symmetric fibers.

Contribution

It provides a complete classification of such Einstein metrics for both isotropy irreducible and product fiber cases, especially for exceptional and classical Lie groups.

Findings

Classified Einstein metrics for isotropy irreducible fibers.

Classified Einstein metrics for product fibers with exceptional G.

Identified orthogonal sum and fiber-restricted Einstein metrics for classical G.

Abstract

We consider a homogeneous fibration $G/L \to G/K$, with symmetric fiber and base, where $G$ is a compact connected semisimple Lie group and $L$ has maximal rank in $G$. We suppose the base space $G/K$ is isotropy irreducible and the fiber $K/L$ is simply connected. We investigate the existence of $G$-invariant Einstein metrics on $G/L$ such that the natural projection onto $G/K$ is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber $K/L$ is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the $G$-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when $G$ is an exceptional Lie group. If $G$ is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.