Seeing Intrinsic Curvature of a Surface Is there any issue with characterizing a surface as being intrinsically flat by saying "at no point does it either resemble a pringle (saddle) or a bullet (elliptic paraboloid)?" I'm trying to develop an eye for seeing the intrinsic curvature in a 2-dimensional surface.
Based on Gauss' initial definition of the intrinsic curvature of a surface at a point $p$ as the product of principal curvatures, $\kappa(p)=\kappa_{min}\kappa_{max}$, I think my characterization is reasonable if informal.
It also seems intuitively telling to take something intrinsically flat like a piece of paper or a placemat and give it some kind of intrinsic curvature only to have it resist by forming kinks and cusps.

I try to describe a 2-D situation verbally, you can look up for descriptive  symbolic equations in a university library.
If you have a brittle egg/bullet  shaped dome $K>0$ and step on it making it flat there will occur radial cracks and gaps upto its disc edge.
On the other hand if you step on a flexible ( thin rubbery) pringle saddle shape $K<0$ and step on it, it becomes flat only after overlapping radial frills and wrinkles develop on the warped flattened disc.
The intrinsic curvature or Gauss curvature $K$ is visibly seen destroyed by forced flattening induced by forces normal to the shell surface.
Non-linear strains are described by pdes of von Kármán in his work on Buckling of doubly curved structures. The theory makes no distinction between radial tension or circumferential compression in creating a dome out of a flat disc. Vice-versa  it makes no distinction between radial compression and circumferential tension in creating a saddle out of a flat disc.
This also due to the fact that it is impossible to isometrically bring pringles and egg shells to a plane without tearing or overlapping by virtue of Gauss Egregium theorem in Surface Theory. Such a mapping is not possible. The curved grid lines will break or the net will fold up.
The theorem states that in pure Bending Gauss curvature $K$ is conserved ( btw as also other items derivable by the first fundamental form of Surface theory ).
So shells or surfaces of constant Gauss curvature will crack up or warp down on being forcibly flattened depending on whether initial Gauss curvature $K$ is positive or negative.
