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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

$$\int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M), \,$$

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 the number 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 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 (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.
Special cases

A number of earlier results in spherical geometry and hyperbolic geometry 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 (inverse) 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
Main article: Descartes' theorem on total angular defect

Descartes' theorem on total angular defect 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 $$\chi=2-2g$$ (where g is the genus, meaning "number of holes"), then the sum of the defect is $$2\pi \chi$$. This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices).

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 M be a finite 2-dimensional pseudo-manifold. Let $$\chi(v)$$ denote the number of triangles containing the vertex v. Then

$$\scriptstyle \sum_{v\in{\mathrm{int}}{M}}(6-\chi(v))+\sum_{v\in\partial M}(4-\chi(v))=6\chi(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.

More specifically, if M is a closed digital 2-dimensional manifold, The genus

$$\scriptstyle g = 1+(M_{5} +2 M_{6}-M_{3})/8, \$$

where M_{i} indicates the set of surface-points each of which has i adjacent points on the surface. See digital topology
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