Megan M. Kerr

Low-dimensional homogeneous Einstein manifolds

We show that compact, simply connected homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics.

NEW HOMOGENEOUS EINSTEIN METRICS OF NEGATIVE RICCI CURVATURE

We construct new homogeneous Einstein spaces with negativeRicci curvature in two ways: First, we give a method for classifying andconstructing a class of rank one Einstein solvmanifolds whose derivedalgebras are two-step nilpotent. As an application, we describe anexplicit continuous family of ten-dimensional Einstein manifolds with atwo-dimensional parameter space, including a continuous subfamily ofmanifolds with negative sectional curvature. Secondly, we obtain newexamples of non-symmetric Einstein solvmanifolds by modifying thealgebraic structure of non-compact irreducible symmetric spaces of rankgreater than one, preserving the (constant) Ricci curvature.

Some New Homogeneous Einstein Metrics on Symmetric Spaces

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank(M)> 1, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup G of Isom(M) acting transitively on M we find all G-homogeneous (non-symmetric) Einstein metrics on M .

A deformation of quaternionic hyperbolic space

We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature, obtained by deforming from the quaternionic hyperbolic space of real dimension 12. We give an explicit description of this family, which is made up of Einstein solvmanifolds which share the same algebraic structure (eigenvalue type) as the rank one symmetric space HH 3 . This deformation includes a continuous family of new homogeneous Einstein spaces with negative sectional curvature.

Homogeneous Einstein–Weyl Structures on Symmetric Spaces

In this paper we examine homogeneous Einstein–Weyl structures and classify them on compact irreducible symmetric spaces. We find that the invariant Einstein–Weyl equation is very restrictive: Einstein–Weyl structures occur only on those spaces for which the isotropy representation has a trivial component, for example, the total space of a circle bundle.

Nonnegatively curved homogeneous metrics in low dimensions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists. In this case, the homogeneous space G/H is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups (H, K, G) whether non-negative curvature is maintained for small deformations. Building on the work of Schwachhöfer and Tapp (J. Geom. Anal. 19(4):929–943, 2009), we examine all G-invariant fibration metrics on G/H for G a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of non-negative sectional curvature.

Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H, K, G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachhöfer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.