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

In this paper, we study the twistor space (Formula presented.) of an oriented Riemannian 4-manifold (Formula presented.) 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 (Formula presented.) forces the underlying manifold (Formula presented.) 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.