A condition for positivity of curvature. (English) Zbl 0777.53036
The authors mention that this paper, minus the application given at the end, was written in 1979 as an addition to the paper of the last two authors [Trans. Am. Math. Soc. 265, 485-493 (1981; Zbl 0465.53028)]. It has never been submitted but now they communicate it because of the revival of interest in the existence of metrics of nonnegative sectional curvature on vector bundles. They consider the following question of S.- T. Yau: does the total space of a bundle over a compact manifold with nonnegative sectional curvature also admit a complete metric with the same property? This is related to O’Neill’s result in Riemannian submersion theory stating that the submersions increase the sectional curvature and to the fact, shown by Cheeger-Gromoll, that if an open manifold has a complete Riemannian metric with nonnegative curvature, then it is the total space of a vector bundle over a compact totally geodesic submanifold.
In this paper the authors consider the case of a principal bundle and provide a necessary and sufficient condition in order that the procedure of the paper mentioned above yields metrics of positive curvature in the total space. It should be noted that for the principal \(G\)-bundle, \(G=S^ 3\), \(SO(3)\) or \(S^ 1\). As an application the authors check which instantons of the Hopf bundle on \(S^ 7\) induce metrics of positive sectional curvature.
In this paper the authors consider the case of a principal bundle and provide a necessary and sufficient condition in order that the procedure of the paper mentioned above yields metrics of positive curvature in the total space. It should be noted that for the principal \(G\)-bundle, \(G=S^ 3\), \(SO(3)\) or \(S^ 1\). As an application the authors check which instantons of the Hopf bundle on \(S^ 7\) induce metrics of positive sectional curvature.
Reviewer: L.Vanhecke (Heverlee)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
Citations:
Zbl 0465.53028References:
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