Gaussian & Mean Curvature

Product and average of the principal curvatures, and the sign of K.

The idea

The Gaussian curvature and the mean curvature condense a surface's bending at a point into single numbers. At each point the surface has two extreme bending rates, the principal curvatures $k_{1}$ and $k_{2}$, and the two standard combinations are $K = k_{1} k_{2} \qquad \text{and} \qquad H = \frac{k_{1} + k_{2}}{2},$ the Gaussian and the mean curvature respectively.

The product and the average behave differently. Reversing the choice of unit normal flips the sign of both principal curvatures, so $H$ changes sign while $K$ does not. The deeper difference is that $K$ is intrinsic: it can be computed from the first fundamental form alone — the fact proved later in this chart as the Theorema Egregium — so bending a surface without stretching it leaves $K$ unchanged. $H$ is extrinsic: rolling a flat sheet into a cylinder changes no internal measurement, and $K$ stays $0$, yet $H$ becomes nonzero.

The sign of $K$ classifies the point. At an elliptic point $K > 0$ and the surface curves the same way in every direction, as on a sphere; at a hyperbolic point $K < 0$ and the surface is saddle-shaped; at a parabolic point $K = 0$, one principal curvature vanishing, as on a cylinder. $H$ governs area instead: a surface with $H = 0$ everywhere is minimal, the shape a soap film settles into.

Ways to work on it

• Walkthrough. Compute K and H from the principal curvatures and classify the point.
• Practice. Compute K or H from given principal curvatures.
• Hardest. Recover the principal curvatures from K and H, then test for minimality.

Not sure where to start? Take the ten-question placement test.