Twistor bundle theory and its application

Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)? J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.

Over an oriented even dimensional Riemannian manifold (M2m, ds2), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive or→ J+(M, ds2) → M. The integrability on an almost complex structure J compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect to J and the canonical almost complex structure J1 on J+(M, ds2), by using moving frame theory. Moreover, for various metrics and a fixed orientation on M, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.