The Curvature of Surfaces

Book 8A has built the language of manifolds: charts, tangent vectors, tensors, forms. This last chapter uses it to answer the question that started differential geometry: how curved is a surface, and can its inhabitants tell? For a surface in space there are two natural answers. One is extrinsic: how fast the surface turns away from its tangent plane, measured by the principal curvatures, the eigenvalues of a symmetric operator. Their product is the Gaussian curvature $K$. The other is intrinsic: what can be measured by someone confined to the surface, who knows only lengths and angles along it.

Gauss's Theorema Egregium (1827), the "remarkable theorem", says the two coincide: $K$, although defined using how the surface sits in space, can be computed from lengths measured within the surface. Curvature is intrinsic. That is why a sphere cannot be flattened without distortion, why a slice of pizza held by its crust stiffens, and why Riemann could define curvature for any manifold with a metric, with no surrounding space at all (Book 9A). The Gauss–Bonnet theorem then ties the total curvature to topology: $\int K\,dA = 2\pi\chi$.

By the end of this chapter you will be able to:

• compute the curvature and torsion of space curves;
• compute the first and second fundamental forms, the shape operator, the principal curvatures, and $K$ and $H$ for a surface;
• state the Theorema Egregium and use it to explain why maps distort and why bent sheets stiffen;
• state Gauss–Bonnet, local and global, and check it on examples;
• compute $K$ for a surface of revolution, $K = -\varphi''/\varphi$, the two-dimensional case of the neck geometry of the Ricci flow.

9.1The pizza slice

9.2Curves in space

A curve $\gamma(s)$ in $\mathbb{R}^3$ parametrised by arc length has unit tangent $T = \gamma'$. Its curvature $\kappa = |T'|$ measures how fast it turns; where $\kappa > 0$ the normal $N = T'/\kappa$ and binormal $B = T\times N$ complete an orthonormal frame, and the Frenet equations

define the torsion $\tau$, which measures how fast the curve twists out of its osculating plane. A circle of radius $r$ has $\kappa = \frac1r$ and $\tau = 0$; a helix has constant $\kappa$ and $\tau$. Curvature and torsion determine a curve up to rigid motion. Plane curves were the subject of the curve shortening flow (6A.8 Curve Shortening and the First Geometric Flows).

9.3The two fundamental forms

Let $S \subset \mathbb{R}^3$ be a surface, locally parametrised by $x(u, v)$, with unit normal $\nu = \frac{x_u\times x_v}{|x_u\times x_v|}$.

The first fundamental form is the metric induced on $S$ by $\mathbb{R}^3$: the restriction of the dot product to tangent planes. In the coordinates $(u, v)$,

a Riemannian metric $g$ with components $g_{11} = E$, $g_{12} = F$, $g_{22} = G$ (8A.7 Tensors and Index Notation). Lengths of curves, angles and areas on $S$ are computed from it: these are the intrinsic quantities.

The shape operator. As one moves along the surface, the normal $\nu$ turns. The shape operator (or Weingarten map) is

the differential of the Gauss map $\nu : S \to S^2$, with a sign. (Since $\nu\cdot\nu = 1$, the derivative of $\nu$ is perpendicular to $\nu$, so it lies in $T_pS$.) It is self-adjoint with respect to $\mathrm I$, and its quadratic form is the second fundamental form

$\mathrm{II}(v, v)$ is the normal curvature of the surface in the direction $v$ (for $|v| = 1$): the curvature of the curve cut out by the plane through $v$ and $\nu$.

Since $\mathcal S$ is self-adjoint, the spectral theorem (1A.6 Symmetric Matrices and the Spectral Theorem) gives real eigenvalues and orthogonal principal directions: the "quadratic forms" thread of the guide. The normal curvature in a direction at angle $\theta$ from the first principal direction is $k_1\cos^2\theta + k_2\sin^2\theta$ (Euler's formula), so $k_1$ and $k_2$ are its maximum and minimum. Changing the choice of normal changes the signs of $k_1$, $k_2$, $H$ but not of $K$. (This guide's $H$ is the sum of the principal curvatures, as in mean curvature flow, 6A.8 Curve Shortening and the First Geometric Flows; some books use the average.)

Examples (Figure 9.1):

• sphere of radius $r$, with the inward normal: $k_1 = k_2 = \frac1r$, $K = \frac{1}{r^2} > 0$, $H = \frac2r$;
• cylinder of radius $r$: $k_1 = \frac1r$ around, $k_2 = 0$ along, $K = 0$, $H = \frac1r$;
• saddle $z = x^2 - y^2$ at the origin: $k_1 = 2$, $k_2 = -2$, $K = -4 < 0$, $H = 0$;
• catenoid, the soap film of 6A.9 Calculus of Variations and Gradient Flows: $H = 0$ everywhere, a minimal surface, and $K < 0$ (Exercise 9.6).

A point with $K > 0$ is elliptic: the surface lies on one side of its tangent plane nearby, like a bowl. $K < 0$ is hyperbolic: the surface crosses its tangent plane like a saddle. $K = 0$ is parabolic (or planar).

9.4The Theorema Egregium

For an orthogonal parametrisation ($F = 0$) there is an explicit formula,

involving only $E$, $G$ and their derivatives (do Carmo, section 4-3). In conformal coordinates, $E = G = e^{2w}$, it reduces to

the formula of 5A.5 Uniformization and the Two-Dimensional Ricci Flow, there taken as a definition and now justified (Exercise 9.7). The proof of the theorem compares two expressions for the third derivatives of the parametrisation: the second fundamental form enters only through $LN - M^2$, and the Gauss equation expresses that combination through the metric alone.

The theorem is astonishing because $K$ was defined by how the normal turns in space, which seems to need the surrounding space. It says an inhabitant of the surface, measuring only lengths along it, can compute $K$. A sheet of paper can be rolled into a cylinder without stretching because both have $K = 0$; it cannot be wrapped around a sphere without crumpling, because a sphere has $K > 0$.

9.5The Gauss–Bonnet theorem

On a surface, a geodesic is a curve that is as straight as possible: its acceleration is normal to the surface (great circles on a sphere). For a geodesic triangle, with interior angles $\alpha$, $\beta$, $\gamma$,

(Gauss, 1827). On a sphere the angles add up to more than $\pi$, on a saddle less, on a plane exactly $\pi$. The triangle with vertices at the North Pole and two points on the Equator a quarter-turn apart has three right angles: excess $\frac\pi2$, which is its area ($\frac18$ of $4\pi$) times $K = 1$ (Figure 9.2).

The total curvature is a topological invariant: however a sphere is deformed, $\int K\,dA = 4\pi$; for a torus it is $0$; for a surface of genus $g$, $2\pi(2 - 2g)$ (7A.3 Manifolds and Surfaces). The proof triangulates the surface into small geodesic triangles, applies the local formula to each, and adds: the angle sums at the vertices give $2\pi V$, the $\pi$'s give $\pi F$, and counting edges gives $\chi$. Descartes' angle-defect theorem for polyhedra (7A.3 Manifolds and Surfaces, last exercise) is the case where all the curvature sits at the vertices.

9.6Surfaces of revolution and necks

Rotate a curve $(\varphi(s), z(s))$, parametrised by arc length ($\varphi'^2 + z'^2 = 1$) with $\varphi > 0$, about the $z$-axis. In the coordinates $(s, \theta)$ the first fundamental form is

and the Theorema Egregium formula gives

(Exercise 9.9). The sphere of radius $r$ has $\varphi = r\sin(s/r)$, so $K = \frac{1}{r^2}$; the cylinder has $\varphi = r$, $K = 0$; a surface whose profile is concave towards the axis, $\varphi'' > 0$, as at the waist of a catenoid or a neck, has $K < 0$. Only the profile function matters, not how the surface sits in space: $K$ is a property of the metric $ds^2 + \varphi^2d\theta^2$.

9.7History

Euler studied the curvature of surfaces through normal sections in the 1760s, and Meusnier and Monge extended his work. Gauss's Disquisitiones generales circa superficies curvas (1827) introduced the Gauss map, proved the Theorema Egregium and the local Gauss–Bonnet theorem for geodesic triangles; Pierre Ossian Bonnet extended it to regions with curved boundary in 1848, and the global form with the Euler characteristic followed with the topology of surfaces later in the century. Frenet (1847) and Serret (1851) found the equations for space curves. Riemann's 1854 lecture took Gauss's intrinsic point of view as the foundation of geometry in any dimension.

9.8Exercises

Solution

$x_\phi = r(-\sin\phi\cos\theta, -\sin\phi\sin\theta, \cos\phi)$, $x_\theta = r(-\cos\phi\sin\theta, \cos\phi\cos\theta, 0)$: $E = r^2$, $F = 0$, $G = r^2\cos^2\phi$. With $\nu = -x/r$: $L = x_{\phi\phi}\cdot\nu = r$, $M = 0$, $N = r\cos^2\phi$. So $K = \frac{LN}{EG} = \frac{r^2\cos^2\phi}{r^4\cos^2\phi} = \frac{1}{r^2}$, and the shape operator is $\frac1r$ times the identity, so $H = \frac2r$.

Solution

The two principal curvatures of a surface of revolution are the curvature of the profile, $\frac{\varphi''}{(1 + \varphi'^2)^{3/2}}$, and $\frac{1}{\varphi\sqrt{1 + \varphi'^2}}$, with opposite signs here (the profile curves away from the axis while the circles curve towards it). With $\varphi = c\cosh(z/c)$: $1 + \varphi'^2 = \cosh^2(z/c)$ and $\varphi'' = \frac1c\cosh(z/c)$, so both have absolute value $\frac{1}{c\cosh^2(z/c)}$. Hence $H = 0$ and $K = -\frac{1}{c^2\cosh^4(z/c)}$.

Solution

$\int_0^{2\pi}\int_0^{2\pi}\cos\phi\,d\theta\,d\phi = 0$. $K > 0$ on the outer half ($|\phi| < \frac\pi2$), $K < 0$ on the inner half facing the hole, and the two contributions cancel exactly.

Solution

$\sqrt{EG} = \varphi$, $E_v = 0$, $G_u = 2\varphi\varphi'$, so $K = -\frac{1}{2\varphi}\big(\frac{2\varphi\varphi'}{\varphi}\big)' = -\frac{\varphi''}{\varphi}$. Sphere: $\varphi'' = -\frac1{r^2}\varphi$, $K = \frac{1}{r^2}$. Cylinder and cone: $\varphi'' = 0$, $K = 0$. $\sinh'' = \sinh$, so $K = -1$.