Data structures for manifold environments

Consider a first-person shooter set not in Euclidean 3-space, but on a curved manifold such as a 3-sphere. By definition of a manifold, space locally around you will look almost like Euclidean space, so it will not be too disorienting to play.

For implementing such a game, what data structure should one use to keep track of where things are? Assume we want something generalizable to many different 3-manifolds, not just the 3-sphere. From a level designer's perspective, we want to be able to create an object and translate it around in space without it weirdly changing shape and size, so straight usage of the 3-sphere equivalent of latitude and longitude will not work. The level designer, and player, should be unaware of any "poles" in the underlying coordinate representation of the manifold.