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arXiv · 1403.4221

On bundles that admit fiberwise hyperbolic dynamics

F. Thomas Farrell·Andrey Gogolev
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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 a number of examples which show that our results are sharp in certain ways or illustrate necessity of various assumptions.

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F. Thomas Farrell, Andrey Gogolev. 2014-07-29. On bundles that admit fiberwise hyperbolic dynamics. https://arxiv.org/abs/1403.4221

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