Euler flows and singular geometric structures

Abstract

Tichler proved in [24] that a manifold admitting a smooth non vanishing and closed one-form bers over a circle. More generally a manifold admitting k independent closed one-forms bers over a torus Tk. In this article we explain a version of this construction for manifolds with boundary using the techniques of b-calculus [18, 13]. 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 ows on manifolds, two dichotomic situations appear. For the rst 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 [19] and what we call b-Beltrami elds is established, thus mimicking the classical correspondence between Beltrami elds and contact structures (see for instance [8]). These results provide a new technique to analyze the geometry of steady uid ows on non-compact manifolds with cylindrical ends.