  We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched
metrics are induced bundles of vector bundles, so their structure groups reduce
from the diffeomorphism group of the sphere to the orthogonal group. This
result implies the existence of many smooth n-sphere bundles over a k-sphere
that do not support strictly 1/4-pinched positively curved Riemannian metrics
on their fibers.
