Gaussian and mean curvature
/ GOW-see-an /
At any point on a surface there are two natural ways to combine the two principal curvatures (the sharpest and gentlest bending you find by looking in different directions): multiply them, or average them. Multiplying gives the Gaussian curvature K = kappa_1 kappa_2; averaging gives the mean curvature H = (kappa_1 + kappa_2)/2. These two numbers summarize, in different ways, how a surface is curved at a point — and they answer very different questions.
The Gaussian curvature K is the product, and its sign alone tells the local shape: K > 0 means both principal curvatures bend the same way, so the surface is dome-like or bowl-like (a sphere, an egg); K < 0 means they bend opposite ways, a saddle (a Pringle, a mountain pass); K = 0 means at least one principal curvature is zero, so the surface is flat in some direction (a plane, a cylinder, a cone). The astonishing fact, Gauss's Theorema Egregium ('remarkable theorem'), is that K is intrinsic — it can be computed purely from the first fundamental form, so a flat ant living on the surface can determine K by measuring distances and angles alone, without ever knowing the surface is embedded in 3-D. The mean curvature H, by contrast, is extrinsic: it depends on how the surface sits in space and flips sign if you flip the normal. H = 0 characterizes minimal surfaces — the shapes soap films take to minimize area.
The split between K and H runs through geometry and physics. K controls intrinsic facts: a triangle on a positively curved surface has angles summing to more than 180 degrees; the Gauss-Bonnet theorem ties the total of K over a surface to its topology (its number of holes), a stunning bridge from local curvature to global shape; and the impossibility of a distortion-free flat map of the Earth is just K of the sphere being nonzero while K of paper is zero. H drives physics of interfaces: surface tension makes a soap film settle at H = 0, and the Young-Laplace law says the pressure difference across a curved liquid surface is proportional to H. A common confusion worth flagging: a cylinder has K = 0 (it is intrinsically flat, you can unroll it) yet H is nonzero (it visibly bends) — proof that the two curvatures are genuinely independent.
A sphere of radius R has both principal curvatures equal to 1/R, so K = 1/R^2 (positive everywhere, same at every point) and H = 1/R. A flat plane has K = 0 and H = 0. A cylinder of radius R has principal curvatures 1/R and 0, giving K = 0 but H = 1/(2R): intrinsically flat like paper, yet extrinsically bent — the cleanest demonstration that K and H measure different things.
The cylinder: K = 0 (unrollable, intrinsically flat) but H nonzero (clearly bent) — the two are independent.
Gaussian curvature K is intrinsic (Theorema Egregium) and survives bending without stretching; mean curvature H is extrinsic and changes when you bend the surface or flip the normal. Confusing the two — for example expecting a cylinder to have nonzero Gaussian curvature because it 'looks curved' — is a classic error.