This is only true for cartesian spaces. If the low-dimensional manifold is "folded" or "crinkly" inside a higher-dimensional space then you can indeed have part

This is only true for cartesian spaces. If the low-dimensional manifold is "folded" or "crinkly" inside a higher-dimensional space then you can indeed have particles that look far apart on the manifold but which are actually close together in the higher-dimensional space. (My terminology here is largely wrong; this is not my field. Perhaps an expert can correct me.)