Application of the Gauss-Bonnet theorem to lensing in the NUT metric
  We show with the help of Fermat's principle that every lightlike geodesic in
the NUT metric projects to a geodesic of a two-dimensional Riemannian metric
which we call the optical metric. The optical metric is defined on a
(coordinate) cone whose opening angle is determined by the impact parameter of
the lightlike geodesic. We show that, surprisingly, the optical metrics on
cones with different opening angles are locally (but not globally) isometric.
With the help of the Gauss-Bonnet theorem we demonstrate that the deflection
angle of a lightlike geodesic is determined by an area integral over the
Gaussian curvature of the optical metric. A similar result is known to be true
for static and spherically symmetric spacetimes. The generalisation to the NUT
spacetime, which is neither static nor spherically symmetric (at least not in
the usual sense), is rather non-trivial.
