Editing Chern–Gauss–Bonnet theorem (section)

== Statement ==
One useful form of the '''Chern theorem''' is that<ref name=":2">{{Cite book|title=Geometry of Differential Forms|volume=201|last=Morita|first=Shigeyuki|date=2001-08-28|publisher=American Mathematical Society|isbn=9780821810453|series=Translations of Mathematical Monographs|location=Providence, Rhode Island|doi=10.1090/mmono/201|url-access=registration|url=https://archive.org/details/geometryofdiffer00mori}}</ref><ref name=":1" />

: <math>\chi(M) = \int_M e(\Omega) </math>

where <math>\chi(M)</math> denotes the [[Euler characteristic]] of ''<math> M </math>.'' The [[Euler class]] is defined as

: <math>e(\Omega) = \frac 1 {(2\pi)^n} \operatorname{Pf}(\Omega).</math>

where we have the [[Pfaffian]] <math>\operatorname{Pf}(\Omega)</math>. Here ''<math> M </math>'' is a [[compact space|compact]] [[Orientability|orientable]] 2''n''-dimensional [[Riemannian manifold]] without [[Boundary (of a manifold)|boundary]], and <math>\Omega</math> is the associated [[curvature form]] of the [[Levi-Civita connection]]. In fact, the statement holds with <math>\Omega</math> the curvature form of any [[metric connection]] on the tangent bundle, as well as for other vector bundles over <math> M </math>.<ref>{{cite journal
| last       = Bell
| first      = Denis
| date       = September 2006
| title      = The Gauss–Bonnet theorem for vector bundles
| journal    = [[Journal of Geometry]]
| volume     = 85
| issue      = 1–2
| pages      = 15–21
| doi        = 10.1007/s00022-006-0037-1
| arxiv= math/0702162
| s2cid = 6856000
}}</ref>

Since the dimension is 2''n'', we have that <math>\Omega</math> is an <math>\mathfrak s\mathfrak o(2n)</math>-valued [[2-form|2-differential form]] on ''<math> M </math>'' (see [[special orthogonal group]]).  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 display="inline">{\bigwedge}^\text{even}\,T^*M</math>. Hence the Pfaffian is a 2''n''-form. It is also an [[invariant polynomial]].

However, Chern's theorem in general is that for any closed <math>C^\infty</math> orientable ''n''-dimensional ''<math> M </math>'',<ref name=":2" />

: <math>\chi(M) = (e(TM), [M]) </math>

where the above pairing (,) denotes the [[cap product]] with the [[Euler class]] of the [[tangent bundle]] <math> TM </math>.

=== Proofs ===
<!-- include Chern's proof, try to provide simple sketch of proof to save space -->
In 1944, the general theorem was first proved by [[S.-S. Chern|S. S. Chern]] in a classic paper published by the [[Princeton University]] math department.<ref>{{Cite journal|last=Chern|first=Shiing-Shen|date=October 1944|title=A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds|url=http://dx.doi.org/10.2307/1969302|journal=The Annals of Mathematics|volume=45|issue=4|pages=747–752|doi=10.2307/1969302|jstor=1969302|issn=0003-486X|url-access=subscription}}</ref>

In 2013, a proof of the theorem via [[Supersymmetry|supersymmetric]] [[Euclidean field theory|Euclidean field theories]] was also found.<ref name=":3" />