There is a static geodesic in which t (time) increases without limit, ignoring possible boundaries or wraparound of position space, and s (its momentum space dual) increases proportionately except that it's periodic. The other coordinates are constant. The principle of relativity states that this geodesic is special only because of the coordinates; physical results would be the same in a moving frame.
In the absence of forces, a geodesic corresponding to uniform motion travels out from an arbitrary (?) origin along a great circle in momentum space, and due to the Christoffel symbols that synchronize space and momentum, it travels out proportionately from an arbitrary origin in position space, in an approximately straight line according to the curvature of the space. Simultaneously it has increasing s and t components which are proportional. When s wraps, momentum space is inverted, and in particular the signs of both the location and the velocity are reversed. Thus the trajectory runs in patches on two great circles. If they pass through the fixed point of the inversion they coincide but with opposite directions.