> Is your point that, for an object to be curved, there must be something for it to curve into? For example, in order for a 2D surface (like the latex in the balloon) to be curved, it necessarily must be in a 3D (or higher) space?
Given the context I'm guessing you asked this rhetorically, but if not: it is not true in general that a curved n-dimensional object must be embedded in a n + k-dimensional space. Curvature can be intrinsic.
>"A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers. An extrinsic curvature, on the other hand, is not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides." (http://mathworld.wolfram.com/IntrinsicCurvature.html) //
but it doesn't make sense to me. A cylinder has no intrinsic curvature but the curvature is discoverable by travelling in one direction only to return from the other?
A sphere has positive Gaussian curvature everywhere, in 3D. You don't need to view it from 4D to discover this, you can draw a triangle on the surface and note that the interior angles add up to >180 degrees. You can't project it into 2D space and keep the curvature, which is why all maps have distortion.
Given the context I'm guessing you asked this rhetorically, but if not: it is not true in general that a curved n-dimensional object must be embedded in a n + k-dimensional space. Curvature can be intrinsic.