Optical geometries and related structures

Abstract

Two natural optical geometries on the space p of all null directions over a four-dimensional Lorentzian manifold M are defined and studied. One of this geometries is never integrable and the other is integrable iff the metric of M is conformally flat. Sections of p forming a zero set of integrability conditions for the latter optical geometry are interpreted as principal null directions on m . Certain well-defined conditions on p are shown to be equivalent to the vanishing of the traceless part of the Ricci tensor of m . Sections of p forming a zero set for these new conditions correspond to the eigendirections of the Ricci tensor of m . An analogy between optical and Hermitian geometries is discussed. Existing (or possible to exist) mutual counterparts between facts from optical and Hermitian geometries are listed. In this analogy, construction of the optical geometries on p constitutes a Lorentzian counterpart of the Atiyah-Hitchin-Singer construction of two natural almost Hermitian structures on the twistor space of four-dimensional Euclidean manifold.