Almost complex manifolds with holomorphic distributions

Abstract

A 2q-dimensional distribution on an orientable manifold M of dimension 2n is called holomorphic if its tangent bundle TM admits a reduction of the structure group to a product U(n − q) × U(q) of two unitary groups. It is shown that a manifold with a holomorphic distribution admits two kinds of metrics g, h, two kinds of almost complex structures J, J!, and an almost product structure Q. The integrability conditions of the structures J, J! and Q are analysed. The (almost) K¨ahler conditions of J, J! with respect to g and h are also treated.

Keywords

• holomorphic distributions
• almost complex structure
• almost product structure
• kähler form
• indefinite almost hermitian metric