2014/03/17 by F. T. Farrell, F. Thomas Farrell, Farrell, F. Thomas +2
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #math.DG #math.DS #math.GT
paper · pdf · doi:10.48550/arxiv.1403.4221
33 pages, 1 figure. We made many presentation improvements in the second version. An overview section was added
openalex publication_date 2014/03/17 · arxiv created 2014/07/29 · arxiv updated 2014/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.