The Geometry of b^k Manifolds

TL;DR

This paper introduces the $b^k$-tangent bundle, extending the geometry of $b$-manifolds, and applies it to classify Poisson structures on surfaces, generalizing existing theorems in the field.

Contribution

It generalizes the $b$-tangent bundle to $b^k$-tangent bundles, describes their geometry, and extends the Mazzeo-Melrose theorem for de Rham theory, with applications to Poisson structures.

Findings

Generalization of $b$-tangent bundles to $b^k$-tangent bundles.

Extension of the Mazzeo-Melrose theorem to $b^k$-manifolds.

Classification of Poisson structures on compact oriented surfaces.

Abstract

Let $Z$ be a hypersurface of a manifold $M$. The $b$-tangent bundle of $(M, Z)$, whose sections are vector fields tangent to $Z$, is used to study pseudodifferential operators and stable Poisson structures on $M$. In this paper we introduce the $b^k$-tangent bundle, whose sections are vector fields with "order $k$ tangency" to $Z$. We describe the geometry of this bundle and its dual, generalize the celebrated Mazzeo-Melrose theorem of the de Rham theory of $b$-manifolds, and apply these tools to classify certain Poisson structures on compact oriented surfaces.