Conditions for Nonnegative Curvature on Vector Bundles and Sphere Bundles

Details Conditions for Nonnegative Curvature on Vector Bundles and Sphere Bundles Conditions for Nonnegative Curvature on Vector Bundles and Sphere Bundles

2002-10-10

arxiv.org/abs/math/0210160v1

Mathematics

Differential Geometry

Scientific paper

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.

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