Optical geometries and related structures (Q1912930)

This article contains a discussion of two optical geometries on the space of all (principal) null directions \(P\) of a four-dimensional Lorentzian manifold \(M\). One of these geometries is always non-integrable, while the other is integrable if and only if \(M\) is conformally flat. Various conditions and properties of \(P\) are analyzed and related to properties on \(M\), and an analogy between optical and Hermitian geometries is discussed. Finally, it is shown that the construction of optical geometries on \(P\) is a Lorentzian analogue of the \textit{M. F. Atiyah}, \textit{N. J. Hitchin} and \textit{I. M. Singer} construction [Proc. R. Soc. Lond., Ser. A 362, 425-461 (1978; Zbl 0389.53011)], of almost Hermitian structures on the twistor space of a four-dimensional Euclidean manifold.