Theorema Egregium

Gauss's Theorema Egregium (Latin for "remarkable theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates of change on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface.

Gauss presented the theorem in this manner (translated from Latin):

Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.

The theorem is "remarkable" because the definition of Gaussian curvature makes ample reference to the specific way the surface is embedded in 3-dimensional space, and it is quite surprising that the result does not depend on its embedding.

In modern mathematical terminology, the theorem may be stated as follows:

A sphere of radius R has constant Gaussian curvature which is equal to 1/R. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances. If one were to step on an empty egg shell, its edges have to split in expansion before being flattened. Mathematically, a sphere and a plane are not isometric, even locally. This fact is significant for cartography: it implies that no planar (flat) map of Earth can be perfect, even for a portion of the Earth's surface. Thus every cartographic projection necessarily distorts at least some distances.

The catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same. Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in other words without extra tension, compression, or shear.

An application of the theorem is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction. This is of practical use in construction, as well as in a common pizza-eating strategy: A flat slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in the direction perpendicular to the fold, an attribute desirable for eating pizza, as it holds its shape long enough to be consumed without a mess. This same principle is used for strengthening in corrugated materials, most familiarly with corrugated fiberboard and corrugated galvanised iron, and in some forms of potato chips as well.

Following Do Carmo we can express the second derivative of a parametrisation of a surface, in terms of the first fundamental form, second fundamental form and Christoffel symbols, then find equations linking the Christoffel symbols to the coefficients of the first fundamental form and their derivatives, showing that these are Christoffel symbols are invariant under isometries. Finally, an equation linking Gaussian curvature to Christoffel symbols shows that it is also invariant under isometries.

Let \(S, \tilde{S}\) be regular surfaces, and let \(\mathbf{r} = \mathbf{r}(u,v)\) be a parametrisation of a patch of the surface \(S\), with unit normal \(\mathbf{N}\). Denote the first derivatives of \(\mathbf{r}\) with respect to \(u\) and \(v\) by \(\mathbf{r}_u\) and \(\mathbf{r}_v\) and the second derivatives by \(\mathbf{r}_{uu}, \mathbf{r}_{uv}, \mathbf{r}_{vv}\). (As our surface is regular, \(\mathbf{r}_{vu} = \mathbf{r}_{uv}\).)

Definition:

A diffeomorphism \(\phi : S \to \tilde{S}\) is an isometry if for all \(p\in S\) and all pairs \(\mathbf{w}_1, \mathbf{w}_2 \in T_p(S)\) the tangent space to \(S\) we have \[\langle \mathbf{w}_1, \mathbf{w}_2 \rangle_p = \langle d\phi_p(\mathbf{w}_1), d\phi_p(\mathbf{w}_1) \rangle_{\phi(p)}.\] In other words, the differential map between tangent spaces, \(d\phi_p:T_p(S)\to T_{\phi(p)}(\tilde{S})\) preserves the inner product.

This definition of isometry applies to the whole surface, for the theorem we only need a weaker definition, defined for small neighbourhoods.

Definition:

A map \(\phi : V \to \tilde{S}\) of a neighbourhood of \(V\) of \(p\in S\) is a local isometry if there a neighbourhood \(\tilde{V}\) of \(\phi(p)\in \tilde{S}\) such that \(\phi:V\to\tilde{V}\) is an isometry.

Condensed: the full section is in Wikipedia.