Astrophysics of Bertrand Space-times
  We construct a model for galactic dark matter that arises as a solution of
Einstein gravity, and is a Bertrand space-time matched with an external
Schwarzschild metric. This model can explain galactic rotation curves. Further,
we study gravitational lensing in these space-times, and in particular we
consider Einstein rings, using the strong lensing formalism of Virbhadra and
Ellis. Our results are in good agreement with observational data, and indicate
that under certain conditions, gravitational lensing effects from galactic dark
matter may be similar to that from Schwarzschild backgrounds.
