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# The Diffeomorphism Type of Canonical Integrations Of Poisson Tensors on Surfaces

Published:2015-05-13
Printed: Sep 2015
• David Martinez-Torres,
PUC-Rio de Janeiro, Departamento de Matemática, Gávea - 22451-900, Rio de Janeiro, Brazil
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## Abstract

A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit canonical integration, $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic to the cotangent bundle $T^*\Sigma$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.
 Keywords: Poisson tensor, Lie groupoid, cotangent bundle
 MSC Classifications: 58H05 - Pseudogroups and differentiable groupoids [See also 22A22, 22E65] 55R10 - Fiber bundles 53D17 - Poisson manifolds; Poisson groupoids and algebroids

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