  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.
