Gaussian Curvature: the Theorema Egregium

Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].

One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name Theorema Egregium or outstanding theorem. In 1828 he published his “Disquisitiones generales circa superficies curvas”, or General investigation of curved surfaces. Gauss defined a quantity that measures the curvature of a two-dimensional surface. He was inspired by his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. The total curvature — or Gaussian curvature — depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used. This is his Theorema Egregium. The Gaussian curvature ${K}$ characterizes the intrinsic geometry of a surface.

What is remarkable about Gauss’s theorem is that the total curvature is an intrinsic quantity. Imagine a geometer living on a two-dimensional surface, or manifold as Riemann called it. The Gaussian curvature can be calculated from measurements that the geometer can make within the surface. It is not necessary to have any information about the higher-dimensional space in which the surface is embedded. This caused the normally restrained Gauss to use the adjective egregium, variously translated as outstanding, exceptional, extraordinary or amazing.

As Lanczos (1970) put it, `In view of his customary reticence, it was an exceptionally jubilant gesture to call one of his theorems “Theorema egregium”. Indeed, his new revelation was so dazzling that he could not but feel that others who claimed to do something important in the field of parallels trailed far behind him.’ The phrase which I have underlined here indicates the close link between the sign of the total curvature and the character of the geometry on the surface.

First and Second Fundamental Forms

Gauss introduced general coordinates ${(u, v)}$ to specify any point on a surface. The cartesian coordinates ${\mathbf{r}=(x, y, z)}$ are functions of these, so that

$\displaystyle x = x(u,v) \,,\qquad y = y(u,v) \,,\qquad z = z(u,v) \,.$

An obvious example is the latitude-longitude coordinate system ${(\phi,\lambda)}$ on the surface of the spherical earth. By generalizing the theorem of Pythagoras, the line element on the surface can be written

$\displaystyle d s^2 = d x^2 + d y^2 + d z^2 = E(u,v) d u^2 + 2 F(u,v) d ud v + G(u,v) dv^2 \ \ (1)$

where the coefficient functions are

$\displaystyle E = \frac{\partial\mathbf{r}}{\partial u}\mathbf{\cdot}\frac{\partial\mathbf{r}}{\partial u} \,, \qquad F = \frac{\partial\mathbf{r}}{\partial u}\mathbf{\cdot}\frac{\partial \mathbf{r}}{\partial v} \,, \qquad G = \frac{\partial\mathbf{r}}{\partial v}\mathbf{\cdot}\frac{\partial\mathbf{r}}{\partial v } \,. \nonumber \ \ \ \ \ (2)$

Equation (1) is known as the first fundamental form of the surface. We note that the functions ${E}$ , ${F}$ and ${G}$ depend only upon values within the surface. They are intrinsic quantities, independent of the space in which the surface is embedded. The first fundamental form (1) contains everything that is required to develop geometry on a two-dimensional surface or manifold. It enables us to calculate the lengths of curves on the surface and the areas of regions on the surface. The geometry may be Euclidean or non-Euclidean, or indeed a combination of these types, depending on the values of the coefficients in (1).

The second fundamental form introduced by Gauss is

$\displaystyle d \sigma^2 = L(u,v) {d u}^{2} + 2 M(u,v){d u}{d v} + N(u,v){d v}^{2}$

where

$\displaystyle L = \mathbf{n\cdot} \frac{\partial^2\mathbf{r}}{\partial u^2} \,, \qquad M = \mathbf{n\cdot} \frac{\partial^2\mathbf{r}}{\partial u\partial v} \,, \qquad N = \mathbf{n\cdot} \frac{\partial^2\mathbf{r}}{\partial v^2} \,.$

The functions ${L}$ , ${M}$ and ${N}$ are extrinsic: they depend upon the embedding space external to the surface, not just on values within the surface itself. Gauss showed that the entire extrinsic geometry of the surface could be characterized by the second fundamental form.

Principal Curvatures and Total Curvature

If a vector ${\mathbf{n}}$ is normal to the surface, any plane containing ${\mathbf{n}}$ cuts the surface in a curve, with curvature ${k}$ . The principal curvatures, ${k_1}$ and ${k_2}$ , are the maximum and minimum values of ${k}$ , and they occur for planes in two orthogonal directions, the principal directions. Gauss focussed on the product of the principal curvatures

$\displaystyle K = k_1 k_2 \,.$

Principal centres of curvature for a
surface of (left) positive curvature and (right) negative curvature.

Gauss showed that ${K}$ could be expressed in terms of the coefficients in the first and second fundamental forms. What we now call the total curvature, or Gaussian curvature, may be defined as

$\displaystyle K(u,v) = \frac{LN-M^2}{EG-F^2}$

The quantities in the numerator depend on values in the 3-dimensional space in which the surface is embedded. The quantities in the denominator are intrinsic quantities.

Gauss then showed that the curvature ${K}$ could be expressed purely in terms of intrinsic quantities. In the case of orthogonal coordinates, for which ${F \equiv 0}$ , his expression is

$\displaystyle K = - \frac{1}{\sqrt{EG}} \left[ \frac{\partial}{\partial u} \left\{ \frac{1}{\sqrt{E}}\frac{\partial \sqrt{G}}{\partial u} \right\} + \frac{\partial}{\partial v} \left\{\frac{1}{\sqrt{G}}\frac{\partial \sqrt{E}}{\partial v} \right\} \right] \ \ \ \ \ (3)$

This may also be written in the form

$\displaystyle K = - \frac{1}{EG}\left[\frac{1}{2}(E_{vv}+G_{uu})- \frac{1}{4}\left({ \frac{E_u G_u + E_v^2}{E} + \frac{E_v G_v + G_u^2}{G} } \right) \right]$

Both of these expressions are intrinsic. The value of ${K}$ determines the nature of the geometry that describes the surface. In the simplest case where ${K}$ is constant, we have Euclidean geometry for ${K=0}$ , elliptic geometry for ${K>0}$ and hyperbolic geometry for ${K<0}$ .

Gauss’s theorem on curvature was made in connection with his development of geometry beyond the limitations of Euclid. It was Gauss who introduced the term non-Euclidean Geometry, although he did not publicize his discovery, fearing that it would meet with a hostile reception. In a letter to Bessel in 1829, He wrote “I fear the cry of the Boetians if I were to voice my views.”

Sources

${\bullet}$ Lanczos, C., 1970: Space Through the Ages. Academic Press, 320pp.

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