Complex product structures and affine foliations

ADRIÁN ANDRADA

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY

SPRINGER

Lugar: Berlin; Año: 2005 vol. 27 p. 377 - 405

0232-704X

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 torsionfree 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.