ABSTRACT
Dimension four is utterly unique from a geometric stand-point. In particular, the orthogonal group is non-simple only for n = 4; and, as a consequence, the Weyl tensor of an oriented Riemannian 4-manifold invariantly splits as a sum of its so-called self-dual and anti-self-dual parts. This decomposition is also conformally invariant. While there are any number of good reasons to study anti-self-dual manifolds, surely one of the most compelling is given by the Penrose twistor correspondence, which associates a complex 3-manifold with every anti-self-dual manifold. This is done in such a way that the conformal metric is completely encoded by the complex structure, and the class of complex 3-manifolds arising in this way can be completely characterized. The chapter also demonstrates constructing new families of anti-self-dual 4-manifolds by smoothing singular quotients of the explicit solutions.
