Gauss–Bonnet theorem

The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.

Statement of the theorem

Suppose $M$ is a compact two-dimensional Riemannian manifold with boundary $partial\; M$. Let $K$ be the Gaussian curvature of $M$, and let $k\_g$ be the geodesic curvature of $partial\; M$. Then :$int\_M\; K;dA+int\_\{partial\; M\}k\_g;ds=2pichi(M),$where "dA" is the element of area of the surface, and "ds" is the line element, along the boundary of "M". Here, $chi(M)$ is the Euler characteristic of $M$.

If the boundary $partial\; M$ is piecewise smooth, then we interpret the integral $int\_\{partial\; M\}k\_g;ds$ as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the angles by which the smooth portions turn at the corners of the boundary.

Interpretation and significance

The theorem applies in particular to compact surfaces without boundary, in which case the integral $int\_\{partial\; M\}k\_g;ds$ can be omitted. It states that the total Gaussian curvature of such a closed surface is equal to 2π times the Euler characteristic of the surface. Note that for orientable compact surfaces without boundary, the Euler characteristic equals $2-2g$, where $g$ is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and $g$ counts thenumber of handles.

If one bends and deforms the surface $M$, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will. The theorem states, somewhat surprisingly, that thetotal integral of all curvatures will remain the same, no matter how the deforming is done. So for instance if you have a sphere with a "dent", then its total curvature is 4π (the Euler characteristic of a sphere being 2), no matter how big or deep the dent.

Compactness of the surface is of crucial importance. Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss-Bonnet formula does not work. It holds true however for the compact closed unit disc, which also has Euler characteristic 1, because of the added boundary integral with value 2π.

As an application, a torus has Euler characteristic 0, so its total curvature must also be zero. If the torus carries the ordinary Riemannian metric from its embedding in R³, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0. It is also possible to construct a torus by identifying opposite sides of a square, in which case the Riemannian metric on the torus is flat and has constant curvature 0, again resulting in total curvature 0. It is not possible to specify a Riemannian metric on the torus with everywhere positive or everywhere negative Gaussian curvature.

The theorem also has interesting consequences for triangles. Suppose "M" is some 2-dimensional Riemannian manifold (not necessarily compact), and we specify a "triangle" on "M" formed by three geodesics. Then we can apply Gauss-Bonnet to the surface "T" formed by the inside of that triangle and the piecewise boundary given by the triangle itself. The geodesic curvature of geodesics being zero, and the Euler characteristic of "T" being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle. Since the turning angle at a corner is equal to π minus the interior angle, we can rephrase this as follows: :The sum of interior angles of a geodesic triangle is equal to π plus the total curvature enclosed by the triangle.In the case of the plane (where the Gaussian curvature is 0 and geodesics are straight lines), we recover the familiar formula for the sum of angles in an ordinary triangle. On the standard sphere, where the curvature is everywhere 1, we see that the angle sum of geodesic triangles is always bigger than π.

Polyhedral analog: Descartes' theorem on total angular defect

Descartes' theorem on total angular defectof a convex polyhedron is the polyhedral analog, and pre-dates Gauss-Bonnet by 200 years.It states that the sum of the defect at all the vertices of a convex polyhedron (i.e., one homeomorphic to the sphere) is 4π.This is the special case of Gauss-Bonnet where the curvature is concentrated at discrete points (the verticies).

Thinking of curvature as a measure, rather than as a function, Descartes' theorem is Gauss-Bonnet where the curvature is a discrete measure, and Gauss-Bonnet for measures generalizes both Gauss-Bonnet for smooth manifolds and Descartes' theorem.

An application: Why does a football/soccer ball have 12 pentagons?

A neat application of the Gauss-Bonnet theorem is the answer to the question: How many pentagons and hexagons does it take to make a football/soccer ball? Assume we use $N$ hexagons and $L$ pentagons; then we have $F=N+L$ faces. Every pentagon (hexagon) has 5 vertices (6 vertices), and each one is shared between 3 faces, hence we have $V=(5\; L+6\; N)/3$ vertices. Similarly, every pentagon (hexagon) has 5 edges (6 edges), and each one is shared between 2 faces, hence we have $E=(5\; L+6\; N)/2$ edges. The Euler characteristic is therefore:$chi(M)=V-E+F=frac\{5\; L+6\; N\}\{3\}-frac\{5\; L+6\; N\}\{2\}+(L+N)=frac\{L\}\{6\}.$

Since the sphere $M$ is compact and has Gaussian curvature $K=1/R^2$, we have

:$int\_M\; K;dA+int\_\{partial\; M\}k\_g;ds=\; frac\{1\}\{R^2\}int\_M\; ;dA+0=4\; pi.$

From the Gauss-Bonnet theorem, $4\; pi=frac\{L\}\{6\}2\; pi$ or $L=12$. The nice result is that we always need 12 pentagons on a football/soccerball, the number of hexagons is in principle unconstrained (but for a real football/soccer ball one obviously chooses a number that makes the ball as spherical as possible). One can also apply this result to Fullerenes.

Combinatorial analog

There are several combinatorial analogs of the Gauss-Bonnet theorem. We state the following one. Let $M$ be a finite 2-dimensional pseudomanifold. Let $chi(v)$ denote the number of triangles containing the vertex $v$. Then:$sum\_\{vindot\{M(6-chi(v))+sum\_\{vinpartial\; M\}(4-chi(v))=6chi(M),$where the first sum ranges over the vertices in the interior of $M$, the second sum is over the boundary vertices, and $chi(M)$ is the Euler characteristic of $M$.

Generalizations

Generalizations of the Gauss-Bonnet theorem to "n"-dimensional Riemannian manifolds were found in the 1940s, by Allendoerfer, Weil, and Chern; see generalized Gauss-Bonnet theorem and Chern-Weil homomorphism. The Riemann-Roch theorem can also be seen as a generalization of Gauss-Bonnet.

An extremely far-reaching generalization of all the above-mentioned theorems is the Atiyah–Singer index theorem.

References

External links

* [http://mathworld.wolfram.com/Gauss-BonnetFormula.html Gauss-Bonnet Theorem] at Wolfram Mathworld