Einstein homogeneous riemannian fibrations
Abstract
This thesis is dedicated to the study of 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 the existence of
Einstein metrics with totally geodesic fibers 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 investigate the existence of such Einstein metrics for invariant bisymmetric
fibrations of maximal rank, i.e., when both the base and the fiber are symmetric
spaces and the base is an isotropy irreducible space of maximal rank. We
find this way new Einstein metrics. 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.