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

The Gauss–Bonnet theorem, or Gauss–Bonnet formula, is an important statement about surfaces in differential geometry, connecting 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.

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.[1]

Statement of the theoremEdit

Suppose   is a compact two-dimensional Riemannian manifold with boundary  . Let   be the Gaussian curvature of  , and let   be the geodesic curvature of  . Then[2][3]

 

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.[2]

Interpretation and significanceEdit

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 (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.

For trianglesEdit

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:[4]

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 casesEdit

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

TrianglesEdit

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.

PolyhedraEdit

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   (where g is the genus, meaning "number of holes"), then the sum of the defect is   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 analogEdit

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 n/(n − 2) and 2n/(n − 2), 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 [5]

 

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.

GeneralizationsEdit

The Chern theorem (after Shiing-Shen Chern 1945) is the 2n-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.

An extremely far-reaching generalization that includes all the above-mentioned theorems is the Atiyah–Singer index theorem, which won both Michael Atiyah and Singer the Abel Prize.

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

In popular cultureEdit

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

See alsoEdit

ReferencesEdit

  1. ^ Chern, Shiing-Shen (March 4, 1998). "Interview with Shiing-Shen Chern" (PDF) (Interview). Interviewed by Allyn Jackson. Retrieved 2019-07-22.
  2. ^ a b do Carmo, Manfredo Perdigão (1992). Riemannian geometry. Boston: Birkhäuser. ISBN 0817634908. OCLC 24667701.
  3. ^ do Carmo, Manfredo Perdigão (1976). Differential geometry of curves and surfaces. Upper Saddle River, N.J.: Prentice-Hall. ISBN 0132125897. OCLC 1529515.
  4. ^ Weeks, Jeffrey R. (2001-12-12). "The Shape of Space". CRC Press. doi:10.1201/9780203912669. ISBN 9780203912669 – via Taylor & Francis. Cite journal requires |journal= (help)
  5. ^ Chen, Li; Rong, Yongwu (August 2010). "Digital topological method for computing genus and the Betti numbers". Topology and its Applications. 157 (12): 1931–1936. doi:10.1016/j.topol.2010.04.006 – via ScienceDirect.

Further readingEdit

External linksEdit