E -Structures and Almost Regular Poisson Manifolds

Abstract In recent years, b -symplectic manifolds have emerged as important objects in symplectic geometry. These manifolds are Poisson manifolds that exhibit symplectic behaviour away from a distinguished hypersurface, where the symplectic form degenerates in a controlled manner. Inspired by this rich landscape, E -structures were introduced by Nest and Tsygan in NT01 as a comprehensive framework for exploring generalizations of b -structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by MS21. We also examine the closely related concept of almost regular Poisson manifolds, as studied in AZ17, which reveals a natural Poisson groupoid associated with these structures. In this article, we investigate the intricate relationship between E -structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the E -structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.