Sphere Bundles with 1/4-pinched Fiberwise Metrics

Thomas Farrell; Zhou Gang; Dan Knopf; Pedro Ontaneda

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arXiv · 1505.03773

Sphere Bundles with 1/4-pinched Fiberwise Metrics

Thomas Farrell·Zhou Gang·Dan Knopf·Pedro Ontaneda

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Abstract

We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not support strictly 1/4-pinched positively curved Riemannian metrics on their fibers.

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Thomas Farrell, Zhou Gang, Dan Knopf, Pedro Ontaneda. 2015-05-14. Sphere Bundles with 1/4-pinched Fiberwise Metrics. https://arxiv.org/abs/1505.03773

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