View source for Gauss-Bonnet theorem

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

<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>

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>

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;

<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>

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

<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>

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).

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]]]:

<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>

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:

<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>

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]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,  , ''Werke'' , '''8''' , K. Gesellschaft Wissenschaft. Göttingen  (1900)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Bonnet,   ''J. École Polytechnique'' , '''19'''  (1848)  pp. 1–146</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.E. Cohn-Vossen,   "Some problems of differential geometry in the large" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C.B. Allendörfer,   A. Weil,   "The Gauss–Bonnet theorem for Riemannian polyhedra"  ''Trans. Amer. Math. Soc.'' , '''53'''  (1943)  pp. 101–129</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Eells,   "A generalization of the Gauss–Bonnet theorem"  ''Trans. Amer. Math. Soc.'' , '''92'''  (1959)  pp. 142–153</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.S. Pontryagin,   "On a connection between homologies and homotopies"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''13'''  (1949)  pp. 193–200  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''5''' , Publish or Perish  (1975)  pp. 1–5</TD></TR></table>