Gauss–Bonnet theorem

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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.

[edit] Statement of the theorem

Suppose M is a compact two-dimensional Riemannian manifold with boundary . Denote by K the Gaussian curvature at points of M, and by k_g the geodesic curvature at points of . Then

where dA is the element of area of the surface, and ds is the line element, along the boundary of M. Here, χ(M) is the Euler characteristic of M.

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.

[edit] 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 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, the difference being the size of the triangle.

[edit] 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.