This paper addresses J. Cheeger and D. Gromoll's question about
which vector bundles admit a complete metric of nonnegative curvature,
and it relates their question to the issue of which sphere bundles
admit a metric of positive curvature. We show that any vector bundle
that 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 M. Strake and G. Walschap's conditions under which a
vector bundle admits a connection metric of nonnegative curvature.
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