gauss–bonnet theorem

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GaussBonnet theorem 1

GaussBonnet theorem

An example of complex region where Gauss-Bonnet theorem can apply. Shows the

sign of geodesic curvature.

The GaussBonnet theorem or

GaussBonnet 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 is a compact two-dimensional

Riemannian manifold with boundary .

Let be the Gaussian curvature of , and let be the geodesic curvature of . Then

where dA is the element of area of the surface, and ds is the line element along the boundary ofM. 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 smoothportions 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

can be omitted. It states that the total Gaussian curvature of such a closed surface is equal to 2 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 4 (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 GaussBonnet 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 2.

As an application, a torus has Euler characteristic 0, so its total curvature must also be zero. If the torus carries theordinary Riemannian metric from its embedding in R

3, then the inside has negative Gaussian curvature, the outside
http://en.wikipedia.org/w/index.php?title=Torushttp://en.wikipedia.org/w/index.php?title=Unit_dischttp://en.wikipedia.org/w/index.php?title=Genus_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Orientable_manifoldhttp://en.wikipedia.org/w/index.php?title=Anglehttp://en.wikipedia.org/w/index.php?title=Piecewise_smoothhttp://en.wikipedia.org/w/index.php?title=Euler_characteristichttp://en.wikipedia.org/w/index.php?title=Volume_elementhttp://en.wikipedia.org/w/index.php?title=Geodesic_curvaturehttp://en.wikipedia.org/w/index.php?title=Gaussian_curvaturehttp://en.wikipedia.org/w/index.php?title=Riemannian_manifoldhttp://en.wikipedia.org/w/index.php?title=Compact_spacehttp://en.wikipedia.org/w/index.php?title=Pierre_Ossian_Bonnethttp://en.wikipedia.org/w/index.php?title=Carl_Friedrich_Gausshttp://en.wikipedia.org/w/index.php?title=Euler_characteristichttp://en.wikipedia.org/w/index.php?title=Curvaturehttp://en.wikipedia.org/w/index.php?title=Surfacehttp://en.wikipedia.org/w/index.php?title=Differential_geometryhttp://en.wikipedia.org/w/index.php?title=Differential_geometryhttp://en.wikipedia.org/w/index.php?title=File%3AGauss-Bonnet_theorem.svg


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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 Mformed by three geodesics. Then we can applyGaussBonnet to the surface Tformed 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 Tbeing 1, the theorem then

states that the sum of the turning angles of the geodesic triangle is equal to 2 minus the total curvature within the

triangle. Since the turning angle at a corner is equal to minus the interior angle, we can rephrase this as follows:

The sum of interior angles of a geodesic triangle is equal to 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 .

Special cases

A number of earlier results in spherical geometry and hyperbolic geometry over the preceding centuries were

subsumed as special cases of GaussBonnet.

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 theoremthe 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 polyhedral 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 g is the genus, meaning "number of holes"), then the sum

of the defect is This is the special case of GaussBonnet, 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 GaussBonnet for measures generalizes both GaussBonnet for smooth

manifolds and Descartes' theorem.
http://en.wikipedia.org/w/index.php?title=Discrete_measurehttp://en.wikipedia.org/w/index.php?title=Measure_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Euler_characteristichttp://en.wikipedia.org/w/index.php?title=Homeomorphichttp://en.wikipedia.org/w/index.php?title=Polyhedronhttp://en.wikipedia.org/w/index.php?title=Descartes%27_theorem_on_total_angular_defecthttp://en.wikipedia.org/w/index.php?title=Johann_Heinrich_Lamberthttp://en.wikipedia.org/w/index.php?title=Hyperbolic_trianglehttp://en.wikipedia.org/w/index.php?title=Girard%27s_theoremhttp://en.wikipedia.org/w/index.php?title=Spherical_trianglehttp://en.wikipedia.org/w/index.php?title=Hyperbolic_trigonometryhttp://en.wikipedia.org/w/index.php?title=Spherical_trigonometryhttp://en.wikipedia.org/w/index.php?title=Geodesic


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Combinatorial analog

There are several combinatorial analogs of the GaussBonnet 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[1]

where indicates the number of surface-points each of which has adjacent points on the surface.

Generalizations

Generalizations of the GaussBonnet theorem to n-dimensional Riemannian manifolds were found in the 1940s, by

Allendoerfer, Weil, and Chern; see generalized GaussBonnet theorem and ChernWeil homomorphism. The

RiemannRoch theorem can also be seen as a generalization of GaussBonnet.

An extremely far-reaching generalization of all the above-mentioned theorems is the AtiyahSinger index theorem.

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

References

[1][1] Chen L and Rong Y, Linear Time Recognition Algorithms for Topological Invariants in 3D, arXiv:0804.1982, ICPR 2008

External links

Hazewinkel, Michiel, ed. (2001), "Gauss-Bonnet theorem" (http://www.encyclopediaofmath.org/index.

php?title=p/g043410),Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

GaussBonnet Theorem (http://mathworld.wolfram.com/Gauss-BonnetFormula.html) at Wolfram Mathworld
http://mathworld.wolfram.com/Gauss-BonnetFormula.htmlhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-1-55608-010-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Springer_Science%2BBusiness_Mediahttp://en.wikipedia.org/w/index.php?title=Encyclopedia_of_Mathematicshttp://www.encyclopediaofmath.org/index.php?title=p/g043410http://www.encyclopediaofmath.org/index.php?title=p/g043410http://en.wikipedia.org/w/index.php?title=Cohn-Vossen%27s_inequalityhttp://en.wikipedia.org/w/index.php?title=Atiyah%E2%80%93Singer_index_theoremhttp://en.wikipedia.org/w/index.php?title=Riemann%E2%80%93Roch_theoremhttp://en.wikipedia.org/w/index.php?title=Chern%E2%80%93Weil_homomorphismhttp://en.wikipedia.org/w/index.php?title=Generalized_Gauss%E2%80%93Bonnet_theoremhttp://en.wikipedia.org/w/index.php?title=Shiing-Shen_Chernhttp://en.wikipedia.org/w/index.php?title=Andr%C3%A9_Weilhttp://en.wikipedia.org/w/index.php?title=Carl_B._Allendoerferhttp://en.wikipedia.org/w/index.php?title=Digital_manifoldhttp://en.wikipedia.org/w/index.php?title=Quadrilateralhttp://en.wikipedia.org/w/index.php?title=Pseudo-manifold


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