Einstein Homogeneous Riemannian Fibrations

TL;DR

This paper investigates the existence and construction of invariant Einstein metrics on homogeneous fibrations with totally geodesic fibers, providing explicit conditions, computations, and new examples in symmetric and bisymmetric spaces.

Contribution

It derives Ricci curvature formulas and necessary conditions for Einstein metrics on such fibrations, and constructs new invariant Einstein metrics on specific classes of homogeneous spaces.

Findings

Existence of new invariant Einstein metrics on bisymmetric fibrations of maximal rank.

Explicit description of isotropy representations and Casimir eigenvalues.

Application to 4-symmetric and Kowalski n-symmetric spaces.

Abstract

In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We show the existence of new invariant Einstein metrics on homogeneous bisymmetric fibrations of maximal rank. For such spaces we describe explicitly the isotropy representation in terms subsets of roots and compute the eigenvalues of the Casimir operators of the fiber along the horizontal direction. Results for compact simply connected 4-symmetric spaces of maximal rank follow from this. Also, new invariant Einstein metrics are found on Kowalski $n$-symmetric spaces.