TITLE: Generalising the Penrose Twistor Fibration
QUESTION [8 upvotes]: As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" generalisation to a fibration of $S^6$ by some projective $n$-space, or possibly flag manifold. What about general $2n$-spheres?

REPLY [19 votes]: Yes, there is such a twistor fibration over each $S^{2n}$, and the resulting manifold is a complex manifold endowed with a holomorphic $n$-plane field transverse to the fibers of the mapping. Namely, one writes $S^{2n} = \mathrm{SO}(2n{+}1)/\mathrm{SO}(2n)$ and then, using the inclusion $\mathrm{U}(n)\subset\mathrm{SO}(2n)$, one has the coset fibration
$$
Z_n = \mathrm{SO}(2n{+}1)/\mathrm{U}(n)\longrightarrow\mathrm{SO}(2n{+}1)/\mathrm{SO}(2n).
$$
The manifold $\mathrm{SO}(2n{+}1)/\mathrm{U}(n)$ canonically has the structure of a complex manifold and is now known as the twistor space of $S^{2n}$.  Interestingly, this space was already used in 1967 by Calabi in his study of minimal $2$-spheres in $S^{2n}$ (Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom. 1 (1967), 111–125), where, using a quite different language, he showed that such immersions all have lifts to the twistor space as holomorphic curves that are tangent to a certain holomorphic plane field on $Z_n$.
Now, there are generalizations of this picture for each of the so-called 'inner' symmetric spaces $G/K$ where $K$ is the fixed subgroup of an involution that is an inner automorphism of $G$.  The twistor fibration is of the form $G/U\to G/K$ where $U\subset K$ is a subgroup such that $K/U$ (the typical fiber of the fibration) is an Hermitian symmetric space.  (There are also other kinds of twistor spaces over $G/K$ that are flag manifolds of the form $G/T$ where $T\subset K$ is a maximal torus.)
For more details on these topics, especially the generalization to the inner symmetric spaces and flag manifolds, one can, for example, have a look at my paper Lie groups and twistor spaces,  Duke Mathematical Journal 52 (1985), pp. 223–261.