绝妙定理

用现代几何语言来说：高斯曲率是规范不变量。

定理叙述

考虑在欧氏空间 R ³ 中的曲面。两点间的内蕴距离定义为在曲面上且连接两点的曲线的长度的最大下界。（例如一个单位球上两个对径点的内蕴距离是π，而在欧氏空间内的距离是2。）连接两点的长度最短的曲线称为测地线。

两个空间若彼此间有保持距离相同的双射，则这两个空间等距。若在空间中每一点都有邻域，可在其上定义如此的双射，则这两个空间是局部等距。

曲面上的一点的高斯曲率有几种定义：

穿过这点的测地线中最大和最小曲率的积，

在这点的无限小的邻域，经高斯映射在单位球面上的像，和在曲面上的这邻域的面积比。

在这些定义中，要算出高斯曲率，先要知道曲率如何嵌入到空间中。而一个曲面的不同嵌入，可得出局部等距的不同曲面。最简单的例子是平面和圆柱体的侧面：将一张摊成平面的纸卷起来包着圆柱体，就得出从平面到圆柱体侧面的局部等距变换，因为这个变形不会改变纸上相近两点的距离。

这条定理称为绝妙定理，是因为高斯曲率的定义需靠曲面到空间的嵌入，而最终结果却不依赖于嵌入。

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:

The Gaussian curvature of a surface is invariant under local isometry.

Elementary applications

A sphere of radius R has constant Gaussian curvature which is equal to 1/R2. 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.

Sketch proof

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,S be regular surfaces,

and let r =r (u,v) be a parametrisation of a patch of the surface S, with unit normal N. Denote the first derivatives

of r with respect to u and v r _(u) and r _(v)

and the second derivatives by r _(uu),r _(uv),r _(vv). (As our surface is regular, r _(vu)=r _(uv).)

Definition:

A diffeomorphism φ:S→ S is an isometry if for all p∈ S and

all pairs w ₁,w ₂∈ Tₚ(S) the tangent space to S we have ⟨ w ₁,w ₂⟩ ₚ=⟨ dφ ₚ(w ₁),dφ ₚ(w ₁)⟩ _(φ (p)).

In other words, the differential map between tangent spaces, dφ ₚ:Tₚ(S)→ T_(φ (p))(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 φ:V→ S of a neighbourhood of V p∈ S
• is a local isometry if there a neighbourhood V φ (p)∈ S

such that φ:V→ V is an isometry.

If there exists local isometries for each p∈ S then S and S

are said to be locally isometric.

An immediate consequence is if x (u,v) is a parametrisation of V⊂ S and φ ∘ x (u,v) is a parametrisation of V⊂ S then the coefficients