Sphere Bundles with 1/4-pinched Fiberwise Metrics

TL;DR

This paper proves that sphere bundles with fiberwise 1/4-pinched metrics are induced by vector bundles, reducing their structure group to the orthogonal group, and shows many such bundles cannot support strictly 1/4-pinched positively curved metrics.

Contribution

It establishes a classification of sphere bundles with 1/4-pinched fiberwise metrics and demonstrates the non-existence of certain positively curved metrics on many such bundles.

Findings

Sphere bundles with 1/4-pinched fiberwise metrics are induced by vector bundles.

Structure group reduces to the orthogonal group for these bundles.

Many sphere bundles over spheres do not admit strictly 1/4-pinched positively curved metrics.

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.