  This paper addresses Cheeger and Gromoll's question of which vector bundles
admit a complete metric of nonnegative curvature, and relates their question to
the issue of which sphere bundles admit a metric of positive curvature. We show
that any vector bundle which admits a metric of nonnegative curvature must
admit a connection, a tensor, and a metric on the base space which together
satisfy a certain differential inequality. On the other hand, a slight
sharpening of this condition is sufficient for the associated sphere bundle to
admit a metric of positive curvature. Our results sharpen and generalize
Walschap and Strake's conditions under which a vector bundle admits a
connection metric of nonnegative curvature.
