For any oriented Riemannian \(4\)-manifold \((M,g)\), the Hodge-\(*\) operator is an involution on \(2\)-forms which induces a decomposition \(\Lambda^ 2=\Lambda^ 2_ +\oplus\Lambda^ 2_ -\) of \(2\)-forms into self-dual and anti-self-dual components depending only on the conformal class \([g]\). The Riemann tensor of \(g\in[g]\) can be viewed as a map \({\mathcal R}: \Lambda^ 2\to\Lambda^ 2\) whose decomposition contains the terms \(C_{\pm}\), \(\varphi\), and \(s\), where \(C_{\pm}\) are the self-dual and anti-self-dual parts of the Weyl tensor, \(\varphi\) is the trace-free Ricci curvature, and \(s\) is the scalar curvature.
In case of Lorentzian signature \((+ + + -)\), the Hodge-\(*\) is not an involution, but in neutral \((+ + - -)\) signature it is, and there is a decomposition exactly as in the Riemannian case, depending on \([g]\), so, anti-selfdual conformal structures exist in neutral signature.
This survey is devoted to study properties of anti-selfdual conformal structures. The authors present relations between ultrahyperbolic differential equations and ASD conformal structures. First, they introduce the local theory of neutral anti-self-dual conformal structures by using spinors and explain how neutral ASD conformal structures are related to Lax pairs and hence integrable systems.
Next, symmetries and twistor theory are discussed to explain the differences between the Riemannian and neutral cases, and to describe various twistor methods of generating neutral ASD conformal structures.
Finally, the authors present some strong global results using a variety of techniques.
For the entire collection see [Zbl 1139.53002].