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.
"Conditions for nonnegative curvature on vector bundles and sphere bundles." Duke Math. J. 116 (1) 77 - 101, 15 January 2003. https://doi.org/10.1215/S0012-7094-03-11613-0