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The total curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434101.png" /> of a two-dimensional compact Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434102.png" />, closed or with boundary, and the rotation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434103.png" /> of its smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434104.png" /> are connected with the Euler characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434105.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434106.png" /> by the relation
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The total curvature $\omega$ of a two-dimensional compact Riemannian manifold $V^2$, closed or with boundary, and the rotation $\tau$ of its smooth boundary $\partial V^2$ are connected with the Euler characteristic $\chi$ of $V^2$ by the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434107.png" /></td> </tr></table>
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$$\omega+\tau=2\pi\chi.$$
  
 
Here
 
Here
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434108.png" /></td> </tr></table>
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$$\omega=\int\limits_{V^2}KdS,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g0434109.png" /> is the Gaussian curvature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341010.png" /> is the area element;
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where $K$ is the Gaussian curvature and $dS$ is the area element;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341011.png" /></td> </tr></table>
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$$\tau=\int\limits_{\partial V^2}k_gdl,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341012.png" /> is the geodesic curvature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341013.png" /> is the line element of the boundary. The Gauss–Bonnet theorem is also valid for a manifold with a piecewise-smooth boundary; in that case
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where $k_g$ is the geodesic curvature and $dl$ is the line element of the boundary. The Gauss–Bonnet theorem is also valid for a manifold with a piecewise-smooth boundary; in that case
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341014.png" /></td> </tr></table>
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$$\tau=\int k_gdl+\sum_i(\pi-\alpha_i),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341015.png" /> is the rotation of the boundary at a vertex. The theorem is valid, in particular, on regular surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341016.png" />. The Gauss–Bonnet theorem was known to C.F. Gauss [[#References|[1]]]; it was published by O. Bonnet [[#References|[2]]] in a special form (for surfaces homeomorphic to a disc).
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where $\pi-\alpha_i$ is the rotation of the boundary at a vertex. The theorem is valid, in particular, on regular surfaces in $E^3$. The Gauss–Bonnet theorem was known to C.F. Gauss [[#References|[1]]]; it was published by O. Bonnet [[#References|[2]]] in a special form (for surfaces homeomorphic to a disc).
  
For a non-compact complete manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341017.png" /> without boundary, an analogue of the Gauss–Bonnet theorem is the Cohn-Vossen inequality [[#References|[3]]]:
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For a non-compact complete manifold $V^2$ without boundary, an analogue of the Gauss–Bonnet theorem is the Cohn-Vossen inequality [[#References|[3]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341018.png" /></td> </tr></table>
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$$\int\limits_{V^2}KdS\leq2\pi\chi.$$
  
 
The Gauss–Bonnet theorem and the given inequality are also valid for convex surfaces and two-dimensional manifolds of bounded curvature.
 
The Gauss–Bonnet theorem and the given inequality are also valid for convex surfaces and two-dimensional manifolds of bounded curvature.
  
The Gauss–Bonnet theorem can be generalized to even-dimensional compact Riemannian manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341019.png" />, closed or with boundary:
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The Gauss–Bonnet theorem can be generalized to even-dimensional compact Riemannian manifolds $V^{2p}$, closed or with boundary:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341020.png" /></td> </tr></table>
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$$\int\limits_{V^{2p}}\Omega dS+\int\limits_{\partial V^{2p}}\phi dl=\frac{(2\pi)^p}{(2p-1)!}\chi,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341022.png" /> denote the volume elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341024.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341025.png" /> is some polynomial in the components of the curvature tensor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341027.png" /> is some polynomial in the components of the curvature tensor and the coefficients of the second fundamental form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043410/g04341028.png" /> [[#References|[4]]]. The Gauss–Bonnet theorem has also been generalized to Riemannian polyhedra [[#References|[5]]]. Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the Riemannian metric [[#References|[4]]], [[#References|[6]]], [[#References|[7]]].
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where $dS$, $dl$ denote the volume elements in $V^{2p}$ and $\partial V^{2p}$, while $\Omega$ is some polynomial in the components of the curvature tensor of $V^{2p}$, and $\phi$ is some polynomial in the components of the curvature tensor and the coefficients of the second fundamental form of $\partial V^{2p}$ [[#References|[4]]]. The Gauss–Bonnet theorem has also been generalized to Riemannian polyhedra [[#References|[5]]]. Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the Riemannian metric [[#References|[4]]], [[#References|[6]]], [[#References|[7]]].
  
 
====References====
 
====References====

Latest revision as of 09:38, 22 August 2014

The total curvature $\omega$ of a two-dimensional compact Riemannian manifold $V^2$, closed or with boundary, and the rotation $\tau$ of its smooth boundary $\partial V^2$ are connected with the Euler characteristic $\chi$ of $V^2$ by the relation

$$\omega+\tau=2\pi\chi.$$

Here

$$\omega=\int\limits_{V^2}KdS,$$

where $K$ is the Gaussian curvature and $dS$ is the area element;

$$\tau=\int\limits_{\partial V^2}k_gdl,$$

where $k_g$ is the geodesic curvature and $dl$ is the line element of the boundary. The Gauss–Bonnet theorem is also valid for a manifold with a piecewise-smooth boundary; in that case

$$\tau=\int k_gdl+\sum_i(\pi-\alpha_i),$$

where $\pi-\alpha_i$ is the rotation of the boundary at a vertex. The theorem is valid, in particular, on regular surfaces in $E^3$. The Gauss–Bonnet theorem was known to C.F. Gauss [1]; it was published by O. Bonnet [2] in a special form (for surfaces homeomorphic to a disc).

For a non-compact complete manifold $V^2$ without boundary, an analogue of the Gauss–Bonnet theorem is the Cohn-Vossen inequality [3]:

$$\int\limits_{V^2}KdS\leq2\pi\chi.$$

The Gauss–Bonnet theorem and the given inequality are also valid for convex surfaces and two-dimensional manifolds of bounded curvature.

The Gauss–Bonnet theorem can be generalized to even-dimensional compact Riemannian manifolds $V^{2p}$, closed or with boundary:

$$\int\limits_{V^{2p}}\Omega dS+\int\limits_{\partial V^{2p}}\phi dl=\frac{(2\pi)^p}{(2p-1)!}\chi,$$

where $dS$, $dl$ denote the volume elements in $V^{2p}$ and $\partial V^{2p}$, while $\Omega$ is some polynomial in the components of the curvature tensor of $V^{2p}$, and $\phi$ is some polynomial in the components of the curvature tensor and the coefficients of the second fundamental form of $\partial V^{2p}$ [4]. The Gauss–Bonnet theorem has also been generalized to Riemannian polyhedra [5]. Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the Riemannian metric [4], [6], [7].

References

[1] C.F. Gauss, , Werke , 8 , K. Gesellschaft Wissenschaft. Göttingen (1900)
[2] O. Bonnet, J. École Polytechnique , 19 (1848) pp. 1–146
[3] S.E. Cohn-Vossen, "Some problems of differential geometry in the large" , Moscow (1959) (In Russian)
[4] V.A. Sharafutdinov, "Relative Euler class and the Gauss–Bonnet theorem" Siberian Math. J. , 14 : 6 (1973) pp. 930–940 Sibirsk Mat. Zh. , 14 : 6 pp. 1321–1635
[5] C.B. Allendörfer, A. Weil, "The Gauss–Bonnet theorem for Riemannian polyhedra" Trans. Amer. Math. Soc. , 53 (1943) pp. 101–129
[6] J. Eells, "A generalization of the Gauss–Bonnet theorem" Trans. Amer. Math. Soc. , 92 (1959) pp. 142–153
[7] L.S. Pontryagin, "On a connection between homologies and homotopies" Izv. Akad. Nauk SSSR Ser. Mat. , 13 (1949) pp. 193–200 (In Russian)


Comments

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 5 , Publish or Perish (1975) pp. 1–5
How to Cite This Entry:
Gauss-Bonnet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss-Bonnet_theorem&oldid=33081
This article was adapted from an original article by Yu.D. Burago (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article