Lower curvature bounds and cohomogeneity one manifolds

AbstractWe 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