Research On the Geometry Related with Structures of Manifolds and Plane Fields

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One of the themes of our research throughout the project was the Lie algebroid which is naturally associated with a 2-vector field. More precisely, let π denote a 2-vector field on M. π is regarded as a bundle map of T*M to TM. The image D of π is a distribution (plane field). Complete integrability of D is investigated by the formulaπ({α,β})=[π(α),π(β)]・1/2 [π,π] (α,β).In fact, D is integrable if and only if Ker π⊂Ker[π,π] holds. Moreover, Ker[π,π] forms a Lie algebroid if it is a subbundle of T*M of constant rank. We generalized this argument on the cotangent bundles to the cases of general Lie algebroids. Further we treated the cases of deformed brackets by closed 1-form.Recently, we showed that the problems are well-treated and proofs are simplified greatly by working in the framework of Dirac structures which was introduced'by Courant and Weinstein before. These results can be found in the following joint papers with K.Mikami.1)K.Mikami and T.Mizutani ;Lie algebroid associated with an almost Dirac structure., Travaux mathematiques Vol 16,255-264(2005)2)K.Mikami and T.Mizutani ;A Lie algebroid and a Dirac structure associated to an almost Dirac structure (To appear)