Twistor Bundles of Foliated Riemannian Manifolds

In this paper, we present an overview of the theory of twistors on foliated manifolds, which has been developed in several papers of ours. We construct the twistor space of the normal bundle of a foliation. We demonstrate that the classical constructions of the twistor theory lead to foliated objects and permit us to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings. Finally, we recall suspension construction, which is used to construct examples of transversely harmonic maps between foliated manifolds and discuss the twistor methods in this case.