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.
You can take a curved object in 3D space and project it into 2D space and it remains curved?
In the examples of Gaussian curvature here https://www.maths.ox.ac.uk/about-us/departmental-art/theory/... they only appear to be curved because they're 3D (an arc-section of a sphere, a paraboloid), none of them is 2D. Are the examples just weak?
Wolfram has a curious definition:
>"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?