Curvature and topology of the optical metric in geometrical optics

We study the geometric structure of refractive index distributions in geometrical optics using the optical metric formulation. Spatial variations of the refractive index generate Gaussian curvature in the optical manifold. Using the Gauss--Bonnet theorem, we relate the integral of this curvature to a global topological invariant. We also introduce a quantity constructed from the refractive index that can be interpreted as a topological index of the refractive medium.