thumb|300px|An example of a complex region where Gauss–Bonnet theorem can apply. Shows the sign of geodesic curvature.

In differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.

In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries.

The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848.

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

:<math>\int_M K\,dA+\int_{\partial M}k_g\,ds=2\pi\chi(M), \, </math>

where is the element of area of the surface, and is the line element along the boundary of . 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.

: The sum of interior angles of a geodesic triangle is equal to plus the total curvature enclosed by the triangle: <math>\sum (\pi - \alpha) = \pi + \int_T K.</math>

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 .

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 (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

Descartes' theorem on total angular defect of a polyhedron is the piecewise-linear analog:

it states that the sum of the defect at all the vertices of a polyhedron which is homeomorphic to the sphere is 4. More generally, if the polyhedron has Euler characteristic (where is the genus, the "number of holes"), then the sum of the defect is .

This is the special case of Gauss–Bonnet in which the curvature is concentrated at discrete points (the vertices). Thinking of curvature as a measure rather than 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

:<math> \sum_{v\,\in\,\operatorname{int}M}\bigl(6 - \chi(v)\bigr) + \sum_{v\,\in\,\partial M}\bigl(3 - \chi(v)\bigr) = 6\chi(M),\ </math>

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 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

:<math> g = 1 + \frac{M_5 + 2 M_6 - M_3}{8}, </math>

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 three-dimensional digital space.

Generalizations

The Chern theorem (after Shiing-Shen Chern 1945) is the -dimensional generalization of GB (also see Chern–Weil homomorphism).

The Riemann–Roch theorem can also be seen as a generalization of GB to complex manifolds.

A far-reaching generalization that includes all the abovementioned theorems is the Atiyah–Singer index theorem.

A generalization to 2-manifolds that need not be compact is Cohn-Vossen's inequality.

thumb|Sculpture made from flat materials using the Gauss–Bonnet Theorem

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.

See also

  • Chern–Gauss–Bonnet theorem
  • Atiyah–Singer index theorem

References

Further reading

  • Gauss–Bonnet Theorem at Wolfram Mathworld