[nine_k]

Thanks, that's informative!

Games on "spherical", or rather, "polar" surfaces are well-known: space flight around a gravitating body, along circular geodesics.

A shooter in a hyperbolic space, and with gravity, could be hilarious.

[zenorogue]

Unfortunately spherical geometry is often done incorrectly (in Civilization the Earth was a cylinder, in other games planets are tori).

What kind of shooter? An Asteroids-like game on hyperbolic manifolds (both 2D and 3D) can be found in HyperRogue, although there is no gravity. Gravity in hyperbolic space is a rather problematic thing, you always get some extremely weird effects, also I believe there are no stable orbits in hyperbolic spaces.