Doubts about higher dimensions and general relativity is common and a crucial point, so I dont think you should get downvoted.<p>Some points which might be helpful. We have a way, using the concept of a manifold, for ants on a surface like a sphere or a dougnut to figure this out without appealing to a third dimension. One could imagining say ant geographers making maps of portions of the surface, and noting how common regions covered in two different maps have different labels&#x2F;coordinates. One can then figure out a definition of when two collections of maps(called an atlas) are equivalent and then show that an atlas for a plane, sphere, doughnut are mutually nonequivalent.<p>But all this is topology and involves global considerations. What is relevant here is local curvature. We can also do this appealing to an extra dimensions. Now, you used the example of a folded paper and you are correct that for an ant on the surface, the curvature is indetectable. The curvature of the paper is extrinsic and not intrinsic. We say that is isometric to flat space, and its curvature tensor is 0.<p>On the other hand, if the ant was on the surface of a ball, it could figure out this curvature intrinsically, for instance, by measuring sum of the angles of a triangle or the distance between parallel lines keeps shrinking. Not only is this intrinsic, but it is locally measurable. One cant have maps, even for a small area of the earth&#x27;s surface, without some kind of distortion because of this intrinsic curvature.<p>An additional complexity - in GR, spacetime is curved rather that just space. Also, dont take &#x27;curvature&#x27; too literally, it is just a way of measuring deviation from numbers that you would get in the flat scenario.<p>For more read up on manifolds, riemann curvature. John Baez had some essays on the geometric meaning of the curvature tensor in terms of the volume of a ball relative to the usual flat Euclidean case.