Duke Mathematical Journal

Conditions for nonnegative curvature on vector bundles and sphere bundles

Kristopher Tapp

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Abstract

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.

Article information

Source
Duke Math. J., Volume 116, Number 1 (2003), 77-101.

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1085598236

Digital Object Identifier
doi:10.1215/S0012-7094-03-11613-0

Mathematical Reviews number (MathSciNet)
MR1950480

Zentralblatt MATH identifier
1044.53026

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Citation

Tapp, Kristopher. Conditions for nonnegative curvature on vector bundles and sphere bundles. Duke Math. J. 116 (2003), no. 1, 77--101. doi:10.1215/S0012-7094-03-11613-0. https://projecteuclid.org/euclid.dmj/1085598236


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