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.
 A: Making this community wiki in the hopes of inviting others to collect and expand on the comments.
Extrinsic and intrinsic geometry Loosely, the "shape" of a smooth surface in Euclidean three-space is visually perceived by its principal curvatures. While these are individually extrinsic (not determined by the first fundamental form/induced metric of the surface), their product, the Gaussian curvature, turns out to be intrinsic (determined by the induced metric).
The shape operator, whose eigenvalues are the principal curvatures, is minus the differential of the Gauss map from the surface to the unit sphere. The determinant is the Jacobian of the Gauss map. In other words, the Gaussian curvature is the "local stretching factor" for how the Gauss map distorts area.
The limits of visualization Whether or not Gaussian curvature can be recognized visually depends on the circumstances. For example, a surface with one large principal curvature, like a thin tube, has wildly differing Gaussian curvature depending on the other (possibly visually indistinguishable from zero) principal curvature.

*

*Specifically, take a point $p$ on a circle of radius greater than $1/r$ (possibly a line), draw a line $\ell$ in the plane of this circle, at distance $1/r^{2}$ from $p$ parallel to the tangent at $p$, and fix an arc $C$ lying to one side of $\ell$. If we revolve $C$ about $\ell$, we obtain a thin tube surface whose Gaussian curvature is between $-1/r$ and $1/r$ (a large interval for small positive $r$), but any two such surfaces are uniformly close (depending on the length of $C$), so visually difficult to distinguish.

Visual recognition requires care even for "surfaces with moderate principal curvatures." For example:

*

*The "top" and "bottom" parallels of a circular torus are parabolic (one principal curvature is $0$).

*The origin of a monkey saddle $z = x^{3} - 3xy^{2}$ is planar (both principal curvatures are $0$).

A: 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.
