Gauss–Bonnet theorem

In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees.

[1] The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries.

[not verified in body] Suppose M is a compact two-dimensional Riemannian manifold with boundary ∂M.

Many standard proofs use the theorem of turning tangents, which states roughly that the winding number of a Jordan curve is exactly ±1.

[2] Suppose M is the northern hemisphere cut out from a sphere of radius R. Its Euler characteristic is 1.

The theorem applies in particular to compact surfaces without boundary, in which case the integral can be omitted.

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.

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.

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.

Here we define a "triangle" on M to be a simply connected region whose boundary consists of three geodesics.

On the standard sphere, where the curvature is everywhere 1, we see that the angle sum of geodesic triangles is always bigger than π.

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.

More generally, if the polyhedron has Euler characteristic χ = 2 − 2g (where g is the genus, the "number of holes"), then the sum of the defect is 2πχ.

This is the special case of Gauss–Bonnet in which the curvature is concentrated at discrete points (the vertices).

Let χ(v) denote the number of triangles containing the vertex v. Then where the first sum ranges over the vertices in the interior of M, the second sum is over the boundary vertices, and χ(M) is the Euler characteristic of M. Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons.

More specifically, if M is a closed 2-dimensional digital manifold, the genus turns out [5] where Mi indicates the number of surface-points each of which has i adjacent points on the surface.

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

The theorem can be used directly as a system to control sculpture - for example, in work by Edmund Harriss in the collection of the University of Arkansas Honors College.