A conformal isothermal reformulation of optical geometry converts the Gauss-Bonnet area term for weak deflection into boundary integrals evaluated on a flat reference ray, reproducing known finite-distance results for Schwarzschild, Reissner-Nordström, and Kottler spacetimes.
Chern, An elementary proof of the existence of isothermal parameters on a surface, Proceedings of the American Mathematical Society6, 771 (1955)
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Boundary-only weak deflection angles from isothermal optical geometry
A conformal isothermal reformulation of optical geometry converts the Gauss-Bonnet area term for weak deflection into boundary integrals evaluated on a flat reference ray, reproducing known finite-distance results for Schwarzschild, Reissner-Nordström, and Kottler spacetimes.