  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.
