On Riemannian 4‐manifolds and their twistor spaces: A moving frame approach

In this paper, we study the twistor space Z$Z$ of an oriented Riemannian 4‐manifold M$M$ using the moving frame approach, focusing, in particular, on the Einstein, non‐self‐dual setting. We prove that any general first‐order linear condition on the almost complex structures of Z$Z$ forces the underlying manifold M$M$ to be self‐dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first‐order quadratic conditions, showing that the Atiyah–Hitchin–Singer almost Hermitian twistor space of an Einstein 4‐manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold.