We experimentally study the impact of intrinsic and extrinsic curvature of space on the evolution of light. We show that the topology of a surface matters for radii of curvature comparable with the wavelength, whereas for macroscopically curved surfaces only intrinsic curvature is relevant. On a surface with constant positive Gaussian curvature we observe periodic refocusing, self-imaging, and diffractionless propagation. In contrast, light spreads exponentially on surfaces with constant negative Gaussian curvature. For the first time we realized two beam interference in negatively curved space.
We consider spatial solitons as, for example, self-confined optical beams in spaces of constant curvature, which are a natural generalization of flat space. Due to the symmetries of these spaces we are able to define respective dynamical parameters, for example, velocity and position. For positively curved space we find stable multiple-hump solitons as a continuation from the linear modes. In the case of negatively curved space we show that no localized solution exists and a bright soliton will always decay through a nonlinear tunneling process
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