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Co-symplectic and co-Kähler structures are the odd-dimensional analogue of symplectic and Kähler structures. In this talk we shall review the basic concepts of both geometries, which are a special case of almost contact metric structures. We will prove a structure theorem for compact co-Kähler manifolds, stating that they are finally covered by the product of a compact Kähler manifold and a circle. This allows us to recover in a very simple way some of the known topological properties of co-Kähler manifolds. As an application, we shall show that in every odd dimension there exists a compact co-Kähler manifold which is not the product of a compact Kähler manifold and a circle. The results are obtained in collaboration with J. Oprea.
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