The Second Fundamental Form, Gaussian Curvature, and Mean Curvature

Core Idea

The second fundamental form measures how a surface bends away from its tangent plane. At each point, the unit normal N changes as you move, and the rate of change defines the shape operator S = -dN, a symmetric linear map on the tangent plane. Its eigenvalues are the principal curvatures k₁ and k₂, the largest and smallest normal curvatures of curves through the point. Their product K = k₁k₂ is the Gaussian curvature and their average H = (k₁ + k₂)/2 is the mean curvature. The sign of K classifies points as elliptic (K > 0), hyperbolic (K < 0) or parabolic (K = 0 with S ≠ 0).

How It's Best Learned

Start with graphs z = f(x, y) at a point where the tangent plane is horizontal; there the second fundamental form is the Hessian of f, and K is its determinant. Compute k₁, k₂, K and H for a sphere, a cylinder and a saddle and connect each to a picture. Then derive the general formulas in terms of the coefficients of the first and second fundamental forms. Finish by stating Gauss's Theorema Egregium, which says K can be computed from the first fundamental form alone.

Common Misconceptions

• A cylinder is curved, so its Gaussian curvature is nonzero. (Its principal curvatures are 1/r and 0, so K = 0; a cylinder can be unrolled flat, which is consistent with K being intrinsic.)
• Mean curvature and Gaussian curvature carry the same information. (A plane and a cylinder both have K = 0 but different H; a saddle-shaped minimal surface has H = 0 but K < 0.)
• Gaussian curvature depends on which unit normal is chosen. (Reversing N changes the sign of both principal curvatures, so H changes sign while K = k₁k₂ does not.)

Explainer

The first fundamental form describes measurement within a surface. It does not see how the surface sits in space: a plane and a cylinder have the same one. To detect bending, follow the unit normal vector N as you move around the surface.

At a point p, let N be the unit normal. For a tangent vector v, the derivative of N in direction v is again tangent to the surface, since N always has length 1. The map S(v) = -dN(v) is the shape operator. It is a symmetric linear map of the tangent plane to itself, so it has two real eigenvalues k₁ ≥ k₂ with perpendicular eigenvectors. These are the principal curvatures, and their eigenvectors give the principal directions.

There is a concrete reading. Take a unit tangent direction v and slice the surface with the plane containing v and N. The slice is a curve, and its curvature, with a sign that tells which way it bends relative to N, is the normal curvature in direction v. As v rotates, the normal curvature varies between k₂ and k₁.

Two combinations are named. The Gaussian curvature is K = k₁k₂ = det S and the mean curvature is H = (k₁ + k₂)/2 = (trace S)/2. In coordinates, with second fundamental form coefficients L = x_uu · N, M = x_uv · N and N₂ = x_vv · N,

K = (L N₂ - M²)/(EG - F²).

For a graph z = f(x, y) at a point where the tangent plane is horizontal, S is the Hessian of f, and K = f_xx f_yy - f_xy².

Examples show what the numbers mean. A sphere of radius R bends equally in every direction: k₁ = k₂ = 1/R, so K = 1/R². A cylinder of radius r bends around but not along its rulings: k₁ = 1/r, k₂ = 0, so K = 0. The saddle z = x² - y² bends up in one direction and down in the other: at the origin k₁ = 2, k₂ = -2, so K = -4.

The sign of K describes the local shape. When K > 0, the surface lies on one side of its tangent plane near p, like a cap. When K < 0, it crosses the tangent plane, like a saddle. Surfaces with H = 0 everywhere are minimal surfaces, the shapes of soap films spanning a wire.

Gauss's Theorema Egregium states that K depends only on E, F and G and their derivatives. So K is intrinsic, even though it was defined using the normal. The Gauss-Bonnet theorem goes further and relates the total Gaussian curvature of a closed surface to its topology.