I'm slightly confused as to how we can use the picture of a 2D surface embedded in 3D space as an analogue to understand (maybe not picture!) 4D spacetime. My initial thinking was that trying to imagine a 4D spacetime isn't really possible, it's just a mathematical concept which one should not try to picture. Am I right in thinking this? But to be a small being/insect on a 2D cylinder, say, on embedded in 3D space the intrinsic geometry would be Euclidean, if the insect traversed the whole cylinder and arrived it at the same point then it could infer some information about the topology of the surface. I'm fine with all this. Problem arises when I try to take these concepts for a 3D manifold embedded in 4D spacetime. If this 3D manifold is curved, then by taking measurements locally we should (like the insect) be able to deduce information about the intrinsic geometry of the space. Does this mean that, if we were near a massive spherical body say, and were able to fly from the north pole, towards the equator, fly to the left (without changing orientation), then fly backwards up to the north pole- the orientation of the spacecraft would differ from the initial? Is this a symptom of 3D space being curved?