Complex Product Structures and Affine Foliations

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 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. Mathematics Subject Classifications (2000): primary: 53C15, 53C07; secondary: 53C50, 53C55.

Однородные Пара-Кэлеровы Многообразия Эйнштейна

“…Основные факты о геометрии пара-гиперкомплексных многообразий собра-ны в [22], где также рассмотрены некоторые примеры.…”

“…В [22]- [26] дано много других интересных примеров ле-воинвариантных гиперсимплектических структур на разрешимых группах Ли. Все такие структуры на 4-мерных группах Ли классифицированы в [24].…”

“…Следующая лемма описывает форму Кошуля ψ ∈ a σ ⊂ g σ ⊂ g, заданную формулой (22), в терминах фундаментальных весов. Доказательство.…”

Complex product structures on 6-dimensional nilpotent Lie algebras

“…Here n is half the dimension of g. If z = z + + z − , with z ± ∈ g ± , then X can be considered as the matrix associated to the endomorphism ( Proof. It has been proved in [1] that Ric is skew-symmetric and that the following relation holds for any x, y ∈ g:…”

“…for any x + , y + ∈ g + , x − , y − ∈ g − , where π ± : g → g ± are the projections (see [1,4]). First, let us state a general result on the torsion-free connection associated to a complex product structure {J, E}, when the product structure E is replaced by another product structure in {cos θ E + sin θ F : θ ∈ [0, 2π)}.…”