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 Schwachh\"ofer 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.Comment: 23 pages; minor revisions, to appear in Geometriae Dedicat

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 nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachh\"ofer and K. Tapp \cite{ST}, 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 nonnegative sectional curvature.Comment: 14 pages, to appear in Annals of Global Analysis and Geometr

Lower curvature bounds and cohomogeneity one manifolds

We shall discuss Riemannian metrics of fixed diameter and controlled lower curvature bound. As in [34], we give a general construction of invariant metrics on homogeneous vector bundles of cohomogeneity one, which implies, in particular, that any cohomogeneity one manifold admits invariant metrics of almost nonnegative sectional curvature. This provides positive evidence for a conjecture by Grove and Ziller [24] which states that any cohomogeneity one manifold should have invariant metrics of nonnegative curvature. © 2002 Elsevier Science B.V. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe