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In [[mathematics]], the '''generalized Gauss–Bonnet theorem''' (''also called [[Shiing-Shen Chern|Chern]]–[[Gauss]]–[[Pierre Ossian Bonnet|Bonnet]] theorem'') presents the [[Euler characteristic]] of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of  the [[Gauss–Bonnet theorem]] (named after [[Carl Friedrich Gauss]] and [[Pierre Ossian Bonnet]]) to higher dimensions.
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Let ''M'' be a [[compact space|compact]] orientable 2''n''-dimensional [[Riemannian manifold]] without boundary, and let <math>\Omega</math> be the [[curvature form]] of the [[Levi-Civita connection]]. This means that <math>\Omega</math> is an <math>\mathfrak s\mathfrak o(2n)</math>-valued [[2-form]] on ''M''. So <math>\Omega</math> can be regarded as a skew-symmetric 2''n'' × 2''n'' matrix whose entries are 2-forms, so it is a matrix over the [[commutative ring]] <math>\bigwedge^{\hbox{even}}T^*M</math>. One may therefore take the [[Pfaffian]] of <math>\Omega</math>,  <math>\mbox{Pf}(\Omega)</math>, which turns out to be a 2''n''-form.
 
The '''generalized Gauss–Bonnet theorem''' states that
:<math>\int_M \mbox{Pf}(\Omega)=(2\pi)^n\chi(M)\ </math>
where <math>\chi(M)</math> denotes the [[Euler characteristic]] of ''M''.
 
==Example: dimension 4==
In dimension <math>n=4</math>, for a compact oriented manifold, we get
 
:<math>\chi(M)=\frac{1}{32\pi^2}\int_M\left(|Rm|^2-4|Rc|^2+R^2\right)d\mu  </math>
 
where <math>Rm</math> is the full [[Riemann curvature tensor]], <math>Rc</math> is the [[Ricci curvature|Ricci curvature tensor]], and <math>R</math> is the [[scalar curvature]].
 
==Further generalizations==
As with the two-dimensional Gauss–Bonnet Theorem, there are generalizations when ''M'' is a [[manifold|manifold with boundary]].
 
The Gauss–Bonnet Theorem can be seen as a special instance in the theory of [[characteristic classes]]. The Gauss–Bonnet integrand is the Euler class.  Since it is a top-dimensional differential form, it is closed.  The naturality of the Euler class means that when you change the Riemannian metric, you stay in the same cohomology class. That means that the integral of the Euler class remains constant as you vary the metric, and so is an invariant of smooth structure.
 
An extremely far-reaching generalization of the Gauss–Bonnet Theorem is the [[Atiyah–Singer index theorem|Atiyah–Singer Index Theorem]]. Let <math>D</math> be a (weakly) [[elliptic differential operator]] between vector bundles. That means that the [[Symbol of a differential operator|principal symbol]] is an isomorphism.  (Strong ellipticity would furthermore require the symbol to be [[positive-definite]].)  Let <math>D*</math> be the [[adjoint operator]].  Then the index is defined as ''dim(ker(D))-dim(ker(D*))'', and by ellipticity is always finite.  The Index Theorem states that this ''analytical index'' is constant as you vary the elliptic operator smoothly. It is in fact equal to a ''topological index'', which can be expressed in terms of characteristic classes.  The 2-dimensional Gauss–Bonnet Theorem arises as the special case where the topological index is defined in terms of [[Betti numbers]] and the analytical index is defined in terms of the Gauss–Bonnet integrand.
 
==See also==
*[[Chern–Weil homomorphism]]
*[[Pontryagin number]]
*[[Pontryagin class]]
 
== References ==
* {{Citation | last1=Cycon  | first1=Hans
            | last2=Froese | first2=Richard
            | last3=Kirsch | first3=Werner
            | last4=Simon  | first4=Barry
            | title=Schroedinger operators
            | publisher=[[Springer-Verlag]]
            | location=Berlin, New York
            | edition=1st | isbn=3-540-16758-7 | year=1987}} Chapter 12
* {{Citation | last1=Chern | first1=Shiing-Shen
            | title=On the curvatura integra in Riemannian manifold
            | journal=[[Annals of Mathematics]]
            | volume=46|issue=4
            | year=1945 |jstor=1969203
            | pages=674–684 }} This is the historically first time that Chern-Gauss-Bonnet was proven without assuming the manifold to be a hypersurface. For hypersurfaces, the result had been shown first by Allendoerfer and Weil in 1940 which is cited in this paper of Chern.
 
{{DEFAULTSORT:Generalized Gauss-Bonnet theorem}}
[[Category:Theorems in differential geometry]]
 
[[de:Satz von Gauß-Bonnet]]

Latest revision as of 11:18, 9 November 2014

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