Sphere bundles with 1/4-pinched fiberwise metrics

Creators: Farrell, Thomas; Gang, Zhou; Knopf, Dan; Ontaneda, Pedro

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 DIFF(S^n) to O(n+1). This result implies the existence of many smooth S^n-bundles over S^k that do not support strictly 1/4-pinched positively curved Riemannian metrics on their fibers.

Additional Information

© 2017 American Mathematical Society. Received by editor(s): April 21, 2016; Received by editor(s) in revised form: June 8, 2016; Published electronically: May 16, 2017. The first and fourth authors thank NSF for support in DMS-1206622. The second author thanks NSF for support in DMS-1308985 and DMS-1443225. The third author thanks NSF for support in DMS-1205270.

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