Light propagation on a two-dimensional curved surface embedded in a three-dimensional space has attracted increasing attention as an analog model of four-dimensional curved spacetime in laboratory. Despite recent developments in modern cosmology on the dynamics and evolution of the universe, investigation of nonlinear dynamics of light in non-Euclidean geometry is still scarce and remains challenging. Here, we study classical and wave chaotic dynamics on a family of surfaces of revolution by considering its equivalent conformally transformed flat billiard, with nonuniform distribution of refractive index. This equivalence is established by showing how these two systems have the same equations and the same dynamics. By exploring the Poincar{e} surface of section, the Lyapunov exponent and the statistics of eigenmodes and eigenfrequency spectrum in the transformed inhomogeneous table billiard, we find that the degree of chaos is fully controlled by a single geometric parameter of the curved surface. A simple interpretation of our findings in transformed billiards, the fictitious force, allows to extend our prediction to other class of curved surfaces. This powerful analogy between two a prior unrelated systems not only brings forward a novel approach to control the degree of chaos, but also provides potentialities for further studies and applications in various fields, such as billiards design, optical fibers, or laser microcavities.
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