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Compatible structures on Lie algebroids and Monge-Ampére operators

Abstract : We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between such composite structures. We then show that the non-degenerate Monge-Ampére structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampére operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold.
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https://hal.archives-ouvertes.fr/hal-00836619
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Submitted on : Friday, June 21, 2013 - 11:35:33 AM
Last modification on : Monday, March 9, 2020 - 6:15:53 PM

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Yvette Kosmann-Schwarzbach, Vladimir Rubtsov. Compatible structures on Lie algebroids and Monge-Ampére operators. Acta Applicandae Mathematicae, Springer Verlag, 2010, 109 (1), pp.101-135. ⟨10.1007/s10440-009-9444-2⟩. ⟨hal-00836619⟩

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