Complex foliations in generalized twistor spaces

Abstract

We consider a natural almost complex distribution on the associated bundle $F^{\left(n\right)}M$ to the principal bundle of the g-orthogonal oriented frames on a Riemannian manifold (M, g), with standard fibre $\frac{SO\left(2n+k\right)}{U\left(n\right)\times SO\left(k\right)}$: we find necessary and sufficient conditions ensuring that the distribution is an almost complex foliation in $F^{\left(n\right)}M$ and we compute the Nijenhuis tensor. Finally, we characterize the local sections of $F^{\left(n\right)}M$.