Twistors and $G$-structures

The authors distinguish a class of twistor spaces $Z=P\times_GS$ that are associated, following Berard-Bergery and Ochiai, with $G$-structures $P$ on even-dimensional manifolds and connections in $P$. It is assumed that $S=G/H$ is a complex totally geodesic submanifold of the affine symmetric space $\operatorname{GL_{2n}}(\mathbf R)/\operatorname{GL_n}(\mathbf C)$. This class includes all the basic examples of twistor spaces fibered over an even-dimensional base. The integrability of the canonical almost complex structure $J_Z$ and the holomorphy of the canonical distribution $\mathscr H_Z$ in $Z$ are studied in terms of some natural $H$-structure with a connection on the manifold $Z$. Some examples are also treated.