Just came across your question, have you found an answer?

I don't know of any specific books but since you mentioned a desire for "very high mathematical rigor," why not just impose some arbitrary metric yourself, from a curved manifold, then solve for the geodesics? The results could be quite interesting depending on what metric is chosen.

You will of course solve the equation for null geodesics:

$${{{d^{\,2}}{q^j}} \over {d{s^2}}} + \Gamma _{k\,l}^j{{{d^{\,2}}{q^k}} \over {d{s^2}}}{{{d^{\,2}}{q^l}} \over {d{s^2}}} = 0$$

where the connection coefficients are calculated from the metric. Any number of generalizations or specializations could be imposed, e.g., Riemannian manifold, non-symmetric connection, etc.

Indeed, you could even cast Fermat's principle in this form.

Note: I added this in the spirit of your post which states that: "... it is also OK to provide an explanation for any sub discipline ..."

This post imported from StackExchange Physics at 2014-03-05 14:45 (UCT), posted by SE-user roybatty