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On bundles that admit fiberwise hyperbolic dynamics

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Abstract

This paper is devoted to rigidity of smooth bundles which are equipped with fiberwise geometric or dynamical structure. We show that the fiberwise associated sphere bundle to a bundle whose leaves are equipped with (continuously varying) metrics of negative curvature is a topologically trivial bundle when either the base space is simply connected or, more generally, when the bundle is fiber homotopically trivial. We present two very different proofs of this result: a geometric proof and a dynamical proof. We also establish a number of rigidity results for bundles which are equipped with fiberwise Anosov dynamical systems. Finally, we present several examples which show that our results are sharp in certain ways or illustrate necessity of various assumptions.

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Acknowledgments

We thank Rafael de la Llave and Federico Rodriguez Hertz for very useful communications. Also we thank the referee for comments which helped to improve our presentation.

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Authors and Affiliations

  1. Binghamton, USA

    F. Thomas Farrell & Andrey Gogolev

Authors
  1. F. Thomas Farrell
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  2. Andrey Gogolev
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Corresponding author

Correspondence to Andrey Gogolev.

Additional information

F. T. Farrell was partially supported by NSF Grant DMS-1206622.

A. Gogolev was partially supported by NSF Grants DMS-1204943, 1266282. He also would like to acknowledge the support provided by Dean’s Research Semester Award at SUNY Binghamton.

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Farrell, F.T., Gogolev, A. On bundles that admit fiberwise hyperbolic dynamics. Math. Ann. 364, 401–438 (2016). https://doi.org/10.1007/s00208-015-1218-8

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  • DOI: https://doi.org/10.1007/s00208-015-1218-8

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