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