  We present shape-preserving spatially accelerating electromagnetic
wavepackets in curved space: wavepackets propagating along non-geodesic
trajectories while recovering their structure periodically. These wavepackets
are solutions to the paraxial and non-paraxial wave equation in curved space.
We analyze the dynamics of such beams propagating on surfaces of revolution,
and find solutions that carry finite power. These solutions propagate along a
variety of non-geodesic trajectories, reflecting the interplay between the
curvature of space and interference effects, with their intensity profile
becoming narrower (or broader) in a scaled self-similar fashion Finally, we
extend this concept to nonlinear accelerating beams in curved space supported
by the Kerr nonlinearity. Our study concentrates on optical settings, but the
underlying concepts directly relate to General Relativity.
