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