The curvature is quite small but is responsible for orbital motion in the steady state condition. I don't think the parent comment was suggesting that space wou

The curvature is quite small but is responsible for orbital motion in the steady state condition. I don't think the parent comment was suggesting that space would "curve into a circle" unless you are implying something else specifically. All ideal orbits, with no other forces involved, are equivalent to geodesics. However that only applies in true free fall. Actual orbits decay due to tiny non-ideal characteristics which have a cumulative effect or stochastic "butterfly effect"-like characteristic over long periods of time. For example in the ideal case, as soon as a satellite fires maneuvering thrusters, it is not following a geodesic for the time that the rockets are active. After the maneuver is completed, a new steady state condition can be calculated piecewise and then it's following a geodesic again. Another example is the ideal state of a satellite being in a low orbit that decays due to friction with the atmosphere, for which there are good estimates but not exact predictions. Due to the friction, there is a tiny acceleration (which happens to become larger over time) and this makes it follow something that starts close to a geodesic but deviates from it.