Abstract
A complex product structure on a manifold is an appropriate combination of a complex structure and a product structure. The existence of such a structure determines many interesting properties of the underlying manifold, notably that the manifold admits a pair of complementary foliations whose leaves carry affine structures. This is due to the existence of a unique torsion-free connection which preserves both the complex and the product structure; this connection is not necessarily flat. We study the existence of complex product structures on tangent bundles of smooth manifolds, and we investigate the structure of manifolds admitting a complex product structure and a compatible hypersymplectic metric, showing that the foliations mentioned earlier are either symplectic or Lagrangian, depending on the symplectic form under consideration.
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Andrada, A. Complex Product Structures and Affine Foliations. Ann Glob Anal Geom 27, 377–405 (2005). https://doi.org/10.1007/s10455-005-3897-y
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DOI: https://doi.org/10.1007/s10455-005-3897-y