Pawel Nurowski
Optical Geometries and Related Structures
The paper is published: J. Geom. and Physics 18 (1996) 335-348

MSC:

53B30 Lorentz metrics, indefinite metrics

53B35 Hermitian and Kahlerian structures, See also {32Cxx}

83C60 Spinor and twistor methods

Abstract: In this work we interpret known facts from the twsitor theory in the
language of optical geometry. \\
Two natural optical geometries on the space $\cal P$ of all null
directions over a 4-dimensional Lorentzian manifold $\cal M$ are
defined and studied. One of this geometries is never integrable and
the other is integrable iff the metric of $\cal M$ is conformally
flat. The sections of $\cal P$ forming a zero set of integrability
conditions for the later optical geometry are interpreted as
principal null directions on $\cal M$.\\
Certain well defined conditions on $\cal P$ are shown to be
equivalent to the vanishing of the traceless part of the Ricci tensor
of $\cal M$. Sections of $\cal P$ forming a zero set for these new
conditions correspond to the eigendirections of the Ricci tensor of
$\cal M$.\\ An analogy between optical and Hermitean geometries is
discussed. Existsting (or possible to exists) mutual counterparts
between facts from optical and Hermitean geometries are listed.

Keywords: space of principal null directions, optical geometries, Hermitian geometries