Twistor bundle theory and its application

Over an oriented even dimensional Riemannian manifold (M2m, ds2), in terms of the Levi-Civita connection form Ω and the canonical form 0 on the bundle of positive or-thonormal frames, we give a detailed description of the twister bundle Гm= SO(2m)/U(m) →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.