Gauss-Bonnet theorem

The Gauss-Bonnet (after Carl Friedrich Gauss and Pierre Ossian Bonnet ) is an important statement on surfaces , their geometry with their topology connects by a relationship between curvature and Euler characteristic is produced. This theorem was found by both mathematicians independently of one another.

While Gauss did not publish his work on this in full (the Disquisitiones circa superficies curvas of 1827 is a special case), the integral formula of Gauß and Bonnet was first published by Bonnet in 1848.

Areas with a smooth edge

statement

Let be a compact and orientable two-dimensional Riemannian manifold with a boundary . Designate with the Gaussian curvature in the points of and with the geodetic curvature of the boundary curve . Then applies ${\displaystyle M}$ ${\displaystyle \partial M}$ ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle k_{g}}$ ${\displaystyle \partial M}$

${\displaystyle \int _{M}K\;dA+\int _{\partial M}k_{g}\;ds=2\pi \chi (M),}$

where is the Euler characteristic of . In particular, the theorem can be applied to manifolds without a boundary. Then the term is dropped. ${\displaystyle \chi (M)}$ ${\displaystyle M}$ ${\displaystyle \textstyle \int _{\partial M}k_{g}\;ds}$

The Gauss-Bonnet theorem can also be generalized to simplicial surfaces, where the angular defect of a corner is defined as a discrete Gaussian curvature.

Explanation of the sentence

If you distort a surface, its Euler characteristic remains unchanged, in contrast to the Gaussian curvature at the individual points. The theorem states that the integral over the curvature, i.e. the total curvature (= integral curvature ), remains unchanged.

A clear interpretation of the integral curvature is obtained by examining the spherical image of an area of the surface . This spherical image is obtained when the normal unit vectors are plotted from the points of the area of the surface from a fixed point, such as the origin of the coordinates. The tips of these vectors then describe an area on the unit sphere which is just the spherical image of the area of . The area of the spherical image is then equal to the sign of the integral of the curvature area of . It is vividly clear that this area increases when the surface curves more strongly. If the area is now bounded by a simple, closed curve , the integral curvature can be expressed as a curve integral over the curve . ${\displaystyle U}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle f}$ ${\displaystyle P}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle U}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle U}$ ${\displaystyle \partial M}$ ${\displaystyle \partial M}$

If you apply the theorem to closed surfaces (see also examples below), you get particularly interesting results. If a closed surface is clearly the surface of a finitely smooth body pierced by holes, then the number is called the gender of the surface ( is the sphere, the torus and the pretzel, ...). The integral curvature of a surface by gender does not depend on the shape of the surface and is the same: ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p=0}$ ${\displaystyle p=1}$ ${\displaystyle p=2}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle p}$

${\displaystyle K(M)=\int _{M}K\;dO=4\pi (1-p)}$

This is an important topological property of the surface with gender , which remains invariant even with any continuous deformation. It also allows the topological properties of a surface to be expressed in terms of differential geometric quantities, in this case the integral curvature. ${\displaystyle p}$

Examples

Hemisphere

• The torus has constant Gaussian curvature and the edge of the torus is the empty set. Therefore the two integral terms are omitted and it follows . Since the gender is the torus and the torus is an orientable surface without a border , equality also applies due to the formula , where gender denotes . ${\displaystyle \mathbb {T} }$ ${\displaystyle 0}$ ${\displaystyle 0=2\pi \chi (\mathbb {T} )}$ ${\displaystyle 1}$ ${\displaystyle 2-2g}$ ${\displaystyle g}$ ${\displaystyle \chi (\mathbb {T} )=0}$
• The round sphere with radius has the Gaussian curvature 1 at every point. The integral over the Gaussian curvature therefore corresponds to its area, which is . On the other hand, the Euler characteristic is that the sphere is obtained as a bond of two (round) surfaces along an edge with a corner (i.e. ). ${\displaystyle M=S^{2}}$ ${\displaystyle 1}$ ${\displaystyle 4\pi }$ ${\displaystyle 2}$ ${\displaystyle 2-1+1=2}$
• The hemisphere with a border is also a surface in the sense of the Gauss-Bonnet theorem. Based on the considerations from the example for the sphere, it becomes clear that the integral over the Gaussian curvature assumes the value in this case . The edge of the hemisphere is just the circle . The geodetic curvature is therefore because it is a measure of the deviation of a curve from a geodetic curve and every curve on the circle must be a geodetic curve. The Euler characteristic is that the hemisphere can be triangulated with a (spherical) triangle . This corresponds to what the Gauss-Bonnet theorem postulates. ${\displaystyle 2\pi }$ ${\displaystyle 0}$ ${\displaystyle 1}$

Areas with a piecewise smooth edge

If the surface has a piecewise differentiable boundary curve, Gauss-Bonnet's theorem can also be formulated for this case. In this case there is an additional term on the left. ${\displaystyle M}$

statement

As before, be a compact and orientable two-dimensional Riemannian manifold with a boundary and let the Gaussian curvature in the points of and with the geodetic curvature of the boundary curve . Then applies ${\displaystyle M}$ ${\displaystyle \partial M}$ ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle k_{g}}$ ${\displaystyle \partial M}$

${\displaystyle \int _{M}K\;dA+\int _{\partial M}k_{g}\;ds+\Sigma AW(\partial M)=2\pi \chi (M).}$

The outer angles are defined as the angles between the right and left limes of the tangential vectors at the kinks of . The boundary curve must be oriented so that it points to the surface. Here is the normal vector of the surface and the tangential vector to the edge curve. ${\displaystyle AW}$ ${\displaystyle \partial M}$ ${\displaystyle N\times c'}$ ${\displaystyle N}$ ${\displaystyle c'}$

example

Rectangle that was triangulated by entering its diagonal d

• A rectangle together with the standard scalar product can be understood as an area with a piecewise smooth edge in the sense of the sentence. In order to determine the Euler characteristic, you can enter a diagonal in the rectangle and you get two triangles. The triangulation now consists of two triangular faces, five edges and four corners. According to the definition of the Euler characteristic, the following applies . Since the rectangle itself and its edge curve have constant curvatures 0, the two integral terms from the set are each 0 and the sum of the outer angles is . Therefore, the statement of the Gauss-Bonnet theorem is reduced to equality in this case . ${\displaystyle M}$ ${\displaystyle \chi (M)=2-5+4=1}$ ${\displaystyle 2\pi }$ ${\displaystyle 2\pi =2\pi }$

Theorema elegantissimum

This conclusion from Gauss says that the total curvature of a simply connected geodesic triangle is equal to its excess angle . For the special case of the 2-sphere, one sees the equivalence to the Gauss-Bonnet theorem using the outer angle sum of an infinitesimal (i.e. flat) triangle of . However, the equivalence also applies in general - in the two-dimensional case - which can be seen with the help of a triangulation , because for it applies: ${\displaystyle \textstyle \int _{\Delta }K\;dA}$ ${\displaystyle 5\pi }$

${\displaystyle 2\pi \chi =2\pi (E-K+F)=2\pi (E-{\frac {3}{2}}F+F)=2\pi E-\pi F=\sum \varepsilon .}$

Gauss-Bonnet-Chern theorem

Gauss-Bonnet's theorem can be generalized to dimensions, which was done by André Weil and Carl B. Allendoerfer in 1943 and with new evidence by Shiing-Shen Chern in 1944. ${\displaystyle n}$

Let be a compact, oriented Riemannian manifold with even dimension and let be the Riemannian curvature tensor . Since it can be used as a vector-valued differential form ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle R(X,Y)=-R(Y,X)}$

${\displaystyle R\in {\mathcal {A}}^{2}(M,{\mathfrak {so}}(TM))}$

be understood. Under these conditions then applies

${\displaystyle \chi (M)={\frac {1}{\sqrt {(2\pi )^{n}}}}\int _{M}\operatorname {Pf} (-R),}$

where is the Pfaff determinant . ${\displaystyle \operatorname {Pf} }$

Knowing that the Fredholm index of equals , where is the outer derivative , this theorem can be understood as a special case of the Atiyah-Singer index theorem . In this context, the Gauss-Bonnet-Chern theorem offers a possibility of calculating the topological index of the operator ${\displaystyle \mathrm {d} +\mathrm {d} ^{*}}$ ${\displaystyle \chi (M)=\operatorname {ind} (\mathrm {d} +\mathrm {d} ^{*})}$ ${\displaystyle \mathrm {d} }$ ${\displaystyle \mathrm {d} +\mathrm {d} ^{*}.}$

See also

• Tightness (differential geometry)

literature

• Manfredo Perdigão do Carmo: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Upper Saddle River NJ 1976, ISBN 0-13-212589-7 .
• John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 .
• Bernhard Schiekel: Curvatures and index sentences - on the trail of Gauß-Bonnet, Cartan, Atiyah-Singer and Witten. An introduction to geometry and topology for physicists. 2nd ed. doi : 10.18725 / OPARU-17162

Individual evidence

1. ↑ Bonnet, Mémoire sur la théorie générale des surfaces, Journal de l'École Polytechnique, Volume 32, 1848, pp. 1-46.
2. ^ Nicole Berline , Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin et al. Springer 1992, ISBN 0-387-53340-0 , p. 33.
3. ^ Nicole Berline, Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin et al. Springer 1992, ISBN 0-387-53340-0 , pp. 149-150.