Gaussian Curvature and Theorema Egregium

We now have two principal curvatures \kappa_1, \kappa_2 at every point of a surface — the sharpest and gentlest bending, from the second fundamental form. Two numbers is one too many for a single verdict on "how curved is it here?", so we combine them. There are two natural combinations, and they turn out to have wildly different personalities.

K = \kappa_1\kappa_2 \quad\text{(Gaussian curvature)}, \qquad H = \frac{\kappa_1 + \kappa_2}{2} \quad\text{(mean curvature)}.

The product K is the star of this page. From the two fundamental forms it can be written

K = \frac{\det \mathrm{II}}{\det \mathrm{I}} = \frac{LN - M^2}{EG - F^2}.

The average H matters too — it governs soap films and minimal surfaces — but it will play the role of the cautionary counter-example at the end.

The sign of K classifies the point

Because K = \kappa_1\kappa_2, its sign is just the sign-rule for a product, and it sorts every point of every surface into one of three types.

K > 0 — elliptic. Both principal curvatures have the same sign, so the surface bends the same way in every direction — it looks like a bowl or a dome. A sphere of radius R has \kappa_1 = \kappa_2 = 1/R, so

K_{\text{sphere}} = \frac{1}{R}\cdot\frac{1}{R} = \frac{1}{R^2} > 0.

K = 0 — parabolic / flat. At least one principal curvature is zero. The plane has \kappa_1=\kappa_2=0; the cylinder has \kappa_1 = 1/R,\ \kappa_2 = 0. Either way K = 0. A cylinder is genuinely flat in this deep sense, even though it plainly bends — a fact that will soon feel less paradoxical.

K < 0 — hyperbolic. The principal curvatures have opposite signs: the surface curves up one way and down the other, a saddle. A Pringle, a mountain pass, the middle of a horse's back — K < 0 everywhere.

See a saddle

Here is the saddle z = x^2 - y^2 (drawn at reduced height so it fits the box). Look at the centre: walk along the x-axis and the surface curves up like a valley floor rising on both sides; walk along the y-axis and it curves down like a ridge falling away. Up one way, down the other — opposite-sign principal curvatures, so K < 0. This is the shape a flat map can never be smoothed onto a sphere, and vice versa. Drag to rotate.

Gauss's Remarkable Theorem

Now the shock. We built K out of the second fundamental form — the object that measures how the surface sits and bends in the surrounding 3D space. You would expect K to be hopelessly extrinsic: change how the surface is embedded and K should change. In 1827 Gauss proved the opposite, and was so pleased he named it the Theorema Egregium — the "Remarkable Theorem".

Read that again: an ant living on the surface, measuring only lengths and angles with the first fundamental form, can determine K without ever glimpsing the third dimension. Curvature — which looked like the most external, "how it's embedded" property imaginable — is secretly built into the internal ruler.

What the Remarkable Theorem forces to be true

You cannot flatten a sphere. A sphere has K = 1/R^2 > 0 everywhere; a flat plane has K = 0. Since an isometry preserves K, no isometry can carry the sphere to the plane. That is the reason every flat map of the Earth must distort something — areas (Mercator's ballooning Greenland), angles, or distances. There is no perfect world map, and Gauss tells us there never can be.

The pizza-slice trick. A flat slice of pizza has K = 0. Fold it into a gentle U across the crust and you have forced one principal curvature to be nonzero — but bending doesn't stretch the dough, so K must stay 0, which pins the other principal curvature at zero. The tip can no longer flop downward. That is why the fold keeps the slice rigid all the way to the tip — you are exploiting Theorema Egregium at lunch.

Developable surfaces. Any surface you can roll out flat without stretching — cylinders, cones, and the crease-free parts of a folded sheet — must have K = 0 at every point, because the flat plane it came from does. These are the developable surfaces, and they are exactly the K = 0 surfaces. It is why a cylinder is "flat" despite its obvious bend: unroll it and it lies down perfectly.

Vignettes

Both tricks are Theorema Egregium in disguise. The Earth's surface has positive Gaussian curvature; paper has zero. Because bending preserves K, you can never lay the round Earth flat without tearing or stretching — so cartographers must choose which distortion to accept. Mercator keeps angles honest (great for navigation) but blows up polar areas until Greenland rivals Africa; equal-area projections keep sizes true but bend every shape.

The pizza fold is the same theorem run the other way. Flat dough has K = 0; folding it into a curl forces one curvature nonzero, so to keep K = 0 the perpendicular curvature is locked at zero — the slice can't droop. Bridges, corrugated roofs and curved potato chips all bank on the same rigidity. One 19th-century theorem, quietly holding your lunch level.

Only K is intrinsic — not the mean curvature H. It is tempting to think "curvature is intrinsic" and lump both together, but H = \tfrac12(\kappa_1+\kappa_2) is thoroughly extrinsic and changes when you bend the surface.

The cleanest example is the paper-to-cylinder bend. A flat plane has \kappa_1=\kappa_2=0, so K=0 and H=0. Roll it (an isometry — no stretching) into a cylinder of radius R: now \kappa_1=1/R,\ \kappa_2=0, so K = 0 is unchanged — as Theorema Egregium demands — but H = 1/(2R) \neq 0 has jumped. The mean curvature felt the bending; the Gaussian curvature did not. When someone says "curvature is preserved by bending", they mean K, and only K.