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We define 2-calibrated structures, which are analogs of symplectic structures in odd dimensions. We show the existence of differential topological constructions compatible with the structure by developing an appropriate approximately holomorphic geometry for 2-calibrated structures.
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David Martínez Torres, The geometry of 2-calibrated manifolds. Port. Math. 66 (2009), no. 4, pp. 427–512DOI 10.4171/PM/1852