Let $ \Phi $ be a $C^2$ codimension one Anosov flow on a compact
Riemannian manifold $M$ of dimension greater than three. Verjovsky
conjectured that $ \Phi $ admits a global cross-section and we affirm
this conjecture when $ \Phi $ is volume preserving in the following
two cases: (1) if the sum of the strong stable and strong
unstable bundle of $\Phi$ is $ \theta $-Hölder continuous for all $
\theta < 1 $; (2) if the center stable bundle of $ \Phi $ is
of class $ C^{1 + \theta} $ for all $ \theta < 1 $. We also show how
certain transitive Anosov flows (those whose center stable bundle is
$C^1$ and transversely orientable) can be ‘synchronized’, that
is,
reparametrized so that the strong unstable determinant of the time $t$
map (for all $t$) of the synchronized flow is identically equal to $
e^t $. Several applications of this method are given, including
vanishing of the Godbillon–Vey class of the center stable foliation of
a codimension one Anosov flow (when $ \dim M > 3 $ and that foliation
is $ C^{1 + \theta} $ for all $ \theta < 1 $), and a positive answer
to a higher-dimensional analog to Problem 10.4 posed by Hurder and
Katok in [HK].
Stable bundle extensions on elliptic Calabi–Yau threefolds
