Some Einstein Homogeneous Riemannian Fibrations

TL;DR

This paper investigates the existence of Einstein metrics on certain homogeneous fibrations, providing necessary conditions, describing special cases, and demonstrating the existence of non-standard Einstein metrics on specific spaces.

Contribution

It generalizes previous results by establishing conditions for Einstein metrics on G-invariant fibrations and constructs new non-standard Einstein metrics on Kowalski spaces.

Findings

Derived necessary conditions using Casimir operators.

Described binormal Einstein metrics as orthogonal sums.

Proved existence of non-standard Einstein metrics on Kowalski spaces.

Abstract

We study the existence of projectable $G$-invariant Einstein metrics on the total space of $G$-equivariant fibrations $M=G/L\to G/K$, for a compact connected semisimple Lie group $G$. We obtain necessary conditions for the existence of such Einstein metrics in terms of appropriate Casimir operators, which is a generalization of the result by Wang and Ziller about Einstein normal metrics. We describe binormal Einstein metrics which are the orthogonal sum of the normal metrics on the fiber and on the base. The special case when the restriction to the fiber and the projection to the base are also Einstein is also considered. As an application, we prove the existence of a non-standard Einstein invariant metric on the Kowalski $n$-symmetric spaces.