Second Fundamental Form and Curvature

Second Fundamental Form: How a Surface Bends in Space

Two Types of Surface Curvature

The first fundamental form describes the "internal life" of a surface — distances for a resident living on it. The second fundamental form describes how the surface bends in the surrounding three-dimensional space — what an external observer sees.

Intuition: a cylinder and a plane have the same "internal" geometry (they can be mapped isometrically onto each other by "unfolding" the cylinder). But they look different from outside: the cylinder is curved in ℝ³, the plane is not. The second form captures this "external" bending.

The Second Quadratic Form

II = −dr · dn = L du² + 2M du dv + N dv²

Coefficients: L = r_{uu} · n, M = r_{uv} · n, N = r_{vv} · n.

Here r_{uu} = ∂²r/∂u², n is the unit normal to the surface. The meaning of L: "how quickly the normal deviates when moving along the u-direction".

Normal curvature of the curve r(u(t), v(t)) on the surface:

κₙ = II / I = (L(u')² + 2Mu'v' + N(v')²) / (E(u')² + 2Fu'v' + G(v')²)

Meusnier’s Theorem: the normal curvature depends only on the direction (u', v'), but not on the particular curve in that direction.

Principal Curvatures and Types of Points

Principal directions — directions in which normal curvature attains extreme values. These are the eigenvectors of the shape matrix (the matrix II relative to I).

Principal curvatures κ₁ and κ₂ — the minimal and maximal normal curvatures.

Calculation: shape matrix Aᵢⱼ = [gᵢₖ]⁻¹ [bᵢₖ], where [gᵢₖ] is the matrix of I, [bᵢₖ] is the matrix of II. Eigenvalues of A are κ₁, κ₂.

Gaussian curvature: K = κ₁ κ₂ = (LN − M²)/(EG − F²)

Mean curvature: H = (κ₁ + κ₂)/2 = (EN − 2FM + GL)/(2(EG − F²))

Types of Points

Elliptic point (K > 0, κ₁ κ₂ same sign): the surface bends in one direction in all directions — "convexity". Example: points on a sphere, ellipsoid, "hill" on a surface.

Hyperbolic point (K < 0, κ₁ and κ₂ have opposite signs): the surface is a "pass" — like a saddle. In some directions it bends upwards, in others downwards.

Parabolic point (K = 0): cylindrical points — one principal curvature is zero. Entire cylinder, cone — K = 0.

Umbilic point (κ₁ = κ₂): curvature is the same in all directions. All points of a sphere are umbilic.

Numeric Examples

Sphere of radius R: κ₁ = κ₂ = 1/R (all points are umbilic). K = 1/R² > 0 (elliptic). H = 1/R.

Cylinder r = R: κ₁ = 1/R (along the generatrix), κ₂ = 0 (along the axis). K = 0. H = 1/(2R).

Saddle z = x² − y² (hyperbolic paraboloid): At point (0,0,0): L = 2, M = 0, N = −2. E = G = 1, F = 0. κ₁ = 2, κ₂ = −2. K = −4 < 0 (hyperbolic). H = 0 (minimal surface!).

A surface with H = 0 is called minimal: it is stationary in area (like a soap film stretched over a wire frame). Soap bubbles are minimal surfaces with H = const.

Real Applications

Architecture: Hyperbolic paraboloid (K < 0) — a strong structure with low weight. Famous roofs: TWA bus station (Saarinen), Sydney Opera House (K < 0 on overhanging sections). Negative Gaussian curvature provides "double" rigidity for the structure.

Computer Graphics: Mean curvature H is used for mesh smoothing: moving vertices along −∇H → minimizing area → smoothing. Umberg–Pollack algorithm.

Medicine: The shape of blood vessels (K and H) is connected with atherosclerosis risk: in points of high curvature (bends) the stress on the vessel wall intensifies.

Curvature in Biology and Materials Science

The second fundamental form describes not only mathematical surfaces but also real physical objects. Cell membranes tend to take a shape with zero mean curvature where possible, and with the minimally possible curvature overall — this minimizes elastic bending energy, described by the Gelfand–Helfrich functional, which is proportional to the integral of square of mean curvature. This principle determines the shape of erythrocytes (biconcave disk), vesicle bubbles and even nuclear membranes. In engineering, thin-walled structures — shells — carry load precisely through curvature: a stone dome supports weight via compressive forces along the surface, not transverse (bending) forces. The design rule: double curvature creates a rigid shell. Single curvature (cylinder) can be easily deformed transversely. That's why an egg is so strong: a biconvex shape distributes the load evenly. In nanotechnology, the curvature of nanoparticle surfaces determines their chemical reactivity and catalytic efficiency — atoms at convex sites have fewer neighbors and are more reactive.

Surface Curvature in Sensor Technologies and Biomechanics

The second fundamental form and the concept of surface curvature have direct technological applications. Microelectromechanical pressure sensors (MEMS) use controlled membrane curvature: deformation of the membrane is proportional to applied pressure, and the mean curvature of the membrane surface is related to stresses by Laplace–Young equations. These principles form the basis of smartphone accelerometers and gyroscopes. In corneal surgery (LASIK laser vision correction) the shape of the optical surface of the eye is described by principal curvatures: the operation changes κ₁ and κ₂ of the cornea, correcting the refractive power of the lens. MRI analysis of brain cortex surfaces through mean and Gaussian curvatures allows detection of anomalies in the development of cortical folds. In aerodynamics, a wing profile is described by the curvature of the median line and thickness, and the lift coefficient is related to the integral curvature of the lower and upper wing surfaces.

Exercise: (a) For the torus r(u,v) = ((R + r cos v)cos u, (R + r cos v)sin u, r sin v): find K and H. At which u, v are the points elliptic/hyperbolic? (b) Prove that for any minimal surface (H = 0) the Gaussian curvature K ≤ 0. (c) Soap film on a flat square contour — what shape does it have? Why is H = 0 associated with soap film?