Sphere bundles with $1/4$-pinched fiberwise metrics
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- by Thomas Farrell, Zhou Gang, Dan Knopf and Pedro Ontaneda PDF
- Trans. Amer. Math. Soc. 369 (2017), 6613-6630 Request permission
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 $\mathrm {DIFF}(\mathbb {S}^n)$ to $\mathrm {O}(n+1)$. This result implies the existence of many smooth $\mathbb {S}^n$-bundles over $\mathbb {S}^k$ that do not support strictly $1/4$-pinched positively curved Riemannian metrics on their fibers.References
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Additional Information
- Thomas Farrell
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, People’s Republic of China
- MR Author ID: 65305
- Email: farrell@math.tsinghua.edu.cn
- Zhou Gang
- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- Address at time of publication: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902
- MR Author ID: 774004
- Email: gzhou@caltech.edu
- Dan Knopf
- Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
- MR Author ID: 661950
- Email: danknopf@math.utexas.edu
- Pedro Ontaneda
- Affiliation: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902
- MR Author ID: 352125
- Email: pedro@math.binghamton.edu
- Received by editor(s): April 21, 2016
- Received by editor(s) in revised form: June 8, 2016
- Published electronically: May 16, 2017
- Additional Notes: 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 - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 6613-6630
- MSC (2010): Primary 57R22; Secondary 53C44
- DOI: https://doi.org/10.1090/tran/6993
- MathSciNet review: 3660235