  Tichler proved that a manifold admitting a smooth closed one-form fibers over
a circle. More generally a manifold admitting $k$ independent closed one-forms
fibers over a torus $T^k$. In this article we explain a version of this
construction for manifolds with boundary using the techniques of $b$-calculus.
We explore new applications of this idea to Fluid Dynamics and more concretely
in the study of stationary solutions of the Euler equations. In the study of
Euler flows on manifolds, two dichotomic situations appear. For the first one,
in which the Bernoulli function is not constant, we provide a new proof of
Arnold's structure theorem and describe $b$-symplectic structures on some of
the singular sets of the Bernoulli function. When the Bernoulli function is
constant, a correspondence between contact structures with singularities and
what we call $b$-Beltrami fields is established, thus mimicking the classical
correspondence between Beltrami fields and contact structures. These results
provide a new technique to analyze the geometry of steady fluid flows on
non-compact manifolds with cylindrical ends.
