Gauss–Bonnet theorem


The Gauss–Bonnet theorem, or Gauss–Bonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their topology. 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.
The simplest case of GB is that the sum of the angles in a planar triangle is 180 degrees. Already one can see the connection between local and global geometry.

Statement

Suppose is a compact two-dimensional Riemannian manifold with boundary. Let be the Gaussian curvature of, and let be the geodesic curvature of. Then
where dA is the element of area of the surface, and ds is the line element along the boundary of M. Here, is the Euler characteristic of.
If the boundary is piecewise smooth, then we interpret the integral 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.
Many standard proofs use the theorem of turning tangents, which states roughly that the winding number of a Jordan curve is exactly ±1.

Interpretation and significance

The theorem applies in particular to compact surfaces without boundary, in which case the integral
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, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.
If one bends and deforms the surface, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will. The theorem states, somewhat surprisingly, that the total 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, 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.

For triangles

Sometimes the GB formula is stated as
where T is a geodesic triangle. That is on M we define a "triangle" formed by three geodesics. We can then apply GB to the surface T formed by the inside of that triangle and the piecewise boundary of the triangle.
The geodesic curvature the bordering geodesics is 0, and the Euler characteristic of T being 1.
Hence 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:
In the case of the plane, 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.

Special cases

A number of earlier results in spherical geometry and hyperbolic geometry, discovered over the preceding centuries, were subsumed as special cases of Gauss–Bonnet.

Triangles

In spherical trigonometry and hyperbolic trigonometry, the area of a triangle is proportional to the amount by which its interior angles fail to add up to 180°, or equivalently by the amount by which its exterior angles fail to add up to 360°.
The area of a spherical triangle is proportional to its excess, by Girard's theorem – the amount by which its interior angles add up to more than 180°, which is equal to the amount by which its exterior angles add up to less than 360°.
The area of a hyperbolic triangle, conversely is proportional to its defect, as established by Johann Heinrich Lambert.

Polyhedra

of a polyhedron is the polyhedral analog:
it states that the sum of the defect at all the vertices of a polyhedron which is homeomorphic to the sphere is 4π. More generally, if the polyhedron has Euler characteristic , then the sum of the defect is
This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points.
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.

Combinatorial analog

There are several combinatorial analogs of the Gauss–Bonnet theorem. We state the following one. Let be a finite 2-dimensional pseudo-manifold. Let denote the number of triangles containing the vertex. Then
where the first sum ranges over the vertices in the interior of, the second sum is over the boundary vertices, and is the Euler characteristic of.
Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons. For polygons of n vertices, we must replace 3 and 6 in the formula above with and, respectively.
For example, for quadrilaterals we must replace 3 and 6 in the formula above with 2 and 4, respectively. More specifically, if is a closed 2-dimensional digital manifold, the genus turns out
where indicates the number of surface-points each of which has adjacent points on the surface. This is the simplest formula of Gauss–Bonnet theorem in 3D digital space.

Generalizations

The Chern theorem is the 2n-dimensional generalization of GB.
The Riemann–Roch theorem can also be seen as a generalization of GB to complex manifolds.
An extremely far-reaching generalization that includes all the above-mentioned theorems is the Atiyah–Singer index theorem, which won both Michael Atiyah and Isadore Singer the Abel Prize.
A generalization to 2-manifolds that need not be compact is the Cohn-Vossen's inequality.

In popular culture

In Greg Egan's novel Diaspora, two characters discuss the derivation of this theorem.