Sphere Bundles with 1/4-pinched Fiberwise Metrics
Geometric Topology
2015-05-15 v1 Differential Geometry
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.
Keywords
Cite
@article{arxiv.1505.03773,
title = {Sphere Bundles with 1/4-pinched Fiberwise Metrics},
author = {Thomas Farrell and Zhou Gang and Dan Knopf and Pedro Ontaneda},
journal= {arXiv preprint arXiv:1505.03773},
year = {2015}
}