Linear and nonlinear optics in curved space

Starting from Maxwell equations in curved space, we derive the field equations of light when constrainted to a curved surface via a possible nonlinear waveguide. We show that surfaces with constant extrinsic curvature, in particular minimal surfaces, allow solutions with a well-defined polarization. We derive a generalized nonlinear Schr\"odinger equation and solve the linear propagation for surfaces of revolution with constant positive and negative Gaussian curvature.