On Riemannian four-manifolds and their twistor spaces: a moving frame\n approach

In this paper we study the twistor space $Z$ of an oriented Riemannian\nfour-manifold $M$ using the moving frame approach, focusing, in particular, on\nthe Einstein, non-self-dual setting. We prove that any general first-order\nlinear condition on the almost complex structures of $Z$ forces the underlying\nmanifold $M$ to be self-dual, also recovering most of the known related\nrigidity results. Thus, we are naturally lead to consider first-order quadratic\nconditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor\nspace of an Einstein four-manifold bears a resemblance, in a suitable sense, to\na nearly K\\"ahler manifold.\n