Gaussian curvature
/ GOWSS-ee-an /
Why can you wrap a tube in a flat sheet of paper but never wrap a basketball without wrinkling the paper? The deep answer is a single number attached to each point of a surface: the Gaussian curvature K. It measures the surface's intrinsic curving — the kind of curving you could detect from inside, by measuring triangles and circles, without ever looking at the surrounding space. A sphere has positive K, a flat plane and a cylinder have K = 0, and a saddle has negative K.
Precisely, the Gaussian curvature at a point is the product of the two principal curvatures: K = k_1 * k_2. In a parametrization it can be computed as K = (e*g - f^2)/(E*G - F^2), the ratio of the second-fundamental-form determinant to the first-fundamental-form determinant. The sign alone classifies the local shape. If K > 0 both principal curvatures have the same sign: the surface is dome-like or bowl-like and lies entirely on one side of its tangent plane (an elliptic point). If K < 0 they have opposite signs: the surface is saddle-shaped and pierces its tangent plane (a hyperbolic point). If K = 0 at least one principal curvature vanishes, like a flat sheet or the wall of a cylinder (a parabolic or planar point). There is also a vivid geometric reading: K is the local area-magnification of the Gauss map, the limiting ratio of normal-image area on the unit sphere to surface area.
Gaussian curvature is the star of the whole field because of Gauss's Theorema Egregium: although K is defined using the principal curvatures (which need the ambient space), it turns out to be computable from the first fundamental form E, F, G alone — it is INTRINSIC. So K survives any bending that does not stretch: rolling paper into a cylinder keeps K = 0 throughout, which is exactly why flat paper takes to a cylinder but rebels against a sphere (whose K = 1/R^2 > 0 cannot be matched by paper's K = 0). It also controls global shape through the Gauss-Bonnet theorem, which equates the total curvature of a closed surface to 2*pi times its Euler characteristic. One firm caution: do not confuse this with the curvature kappa of a one-dimensional curve, which is extrinsic and unrelated; Gaussian curvature is a property of a two-dimensional surface, and its intrinsic nature is its whole point.
A sphere of radius R has both principal curvatures equal to 1/R, so K = (1/R)(1/R) = 1/R^2 > 0 at every point: a bigger sphere is less curved, a smaller one more, just as intuition expects. A flat plane has K = 0. A cylinder also has K = 0, because one of its principal curvatures (along the axis) is 0, killing the product even though the surface visibly curves — that vanishing K is the precise statement that a cylinder is 'really' flat and unrolls onto paper. A saddle surface like z = x^2 - y^2 has K < 0 at the centre.
Sphere K = 1/R^2 > 0; plane and cylinder K = 0; saddle K < 0.
K = 0 does NOT mean 'visually flat.' A cylinder is visibly curved yet has K = 0 everywhere, because one principal curvature is zero and Gaussian curvature is the PRODUCT. K = 0 means intrinsically flat — unrollable onto a plane without stretching — which is a statement about the metric, not about looking flat.