Editing Chern–Gauss–Bonnet theorem

{{short description|Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature}}
In [[mathematics]], the '''Chern theorem''' (or the '''Chern–Gauss–Bonnet theorem'''<ref>{{cite arXiv|last1=Gilkey|first1=P.|last2=Park|first2=J. H.|date=2014-09-16|title=A proof of the Chern-Gauss-Bonnet theorem for indefinite signature metrics using analytic continuation|class=math.DG|eprint=1405.7613}}</ref><ref>{{Cite journal|last1=Buzano|first1=Reto|last2=Nguyen|first2=Huy The|date=2019-04-01|title=The Higher-Dimensional Chern–Gauss–Bonnet Formula for Singular Conformally Flat Manifolds|journal=The Journal of Geometric Analysis|language=en|volume=29|issue=2|pages=1043–1074|doi=10.1007/s12220-018-0029-z|issn=1559-002X|doi-access=free|hdl=2318/1701050|hdl-access=free}}</ref><ref name=":3">{{cite arXiv|last=Berwick-Evans|first=Daniel|date=2013-10-20|title=The Chern-Gauss-Bonnet Theorem via supersymmetric Euclidean field theories|class=math.AT|eprint=1310.5383}}</ref> after [[Shiing-Shen Chern]], [[Carl Friedrich Gauss]], and [[Pierre Ossian Bonnet]]) states that the [[Euler–Poincaré characteristic]] (a [[topological invariant]] defined as the alternating sum of the [[Betti number]]s of a [[topological space]]) of a [[closed manifold|closed]] even-dimensional [[Riemannian manifold]] is equal to the [[integral]] of a certain polynomial (the [[Euler class]]) of its [[Curvature of Riemannian manifolds|curvature form]] (an [[analytical invariant]]).

It is a highly non-trivial generalization of the classic [[Gauss–Bonnet theorem]] (for 2-dimensional manifolds / [[Surface (mathematics)|surfaces]]) to higher even-dimensional Riemannian manifolds. In 1943, [[Carl B. Allendoerfer]] and [[André Weil]] proved a special case for extrinsic manifolds. In a classic paper published in 1944, [[Shiing-Shen Chern]] proved the theorem in full generality connecting global [[topology]] with local [[geometry]].<ref name=":0">{{Cite journal|last=Chern|first=Shiing-shen|date=October 1945|title=On the Curvatura Integra in a Riemannian Manifold|journal=[[The Annals of Mathematics]]|volume=46|issue=4|pages=674–684|doi=10.2307/1969203|jstor=1969203|s2cid=123348816 }}</ref>

The [[Riemann–Roch theorem]] and the [[Atiyah–Singer index theorem]] are other generalizations of the Gauss–Bonnet theorem.

== 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" />

== Applications ==
The Chern–Gauss–Bonnet theorem can be seen as a special instance in the theory of [[characteristic classes]]. The Chern 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 changing the [[Riemannian metric]], one stays in the same [[cohomology class]]. That means that the integral of the Euler class remains constant as the metric is varied and is thus a global invariant of the smooth structure.<ref name=":1">{{Cite book|title=Schrödinger operators, with applications to quantum mechanics and global geometry|date=1987|publisher=Springer-Verlag|others=Cycon, H. L. (Hans Ludwig), 1942-, Simon, Barry, 1946-, Beiglböck, E., 1939-|isbn=978-0387167589|location=Berlin|oclc=13793017}}</ref>

The theorem has also found numerous applications in [[physics]], including:<ref name=":1" />

* [[Geometric phase|adiabatic phase]] or [[Berry's phase]], 
* [[string theory]], 
* [[condensed matter physics]],
* [[topological quantum field theory]],
* [[topological phases of matter]] (see the 2016 Nobel Prize in physics by [[Duncan Haldane]] et al.).

== Special cases ==

===Four-dimensional manifolds===
In dimension <math>2n=4</math>, for a compact oriented manifold, we get

:<math>\chi(M) = \frac{1}{32\pi^2} \int_M \left( |\text{Riem}|^2 - 4 |\text{Ric}|^2 + R^2 \right) \, d\mu  </math>

where <math>\text{Riem}</math> is the full [[Riemann curvature tensor]], <math>\text{Ric}</math> is the [[Ricci curvature|Ricci curvature tensor]], and <math>R</math> is the [[scalar curvature]]. This is particularly important in [[general relativity]], where spacetime is viewed as a 4-dimensional manifold. 

In terms of the orthogonal [[Ricci decomposition]] of the Riemann curvature tensor, this formula can also be written as

:<math>\chi(M) = \frac{1}{8\pi^2} \int_M \left( \frac{1}{4}|W|^2 - \frac{1}{2} |Z|^2 + \frac{1}{24}R^2 \right) \, d\mu  </math>

where <math>W</math> is the [[Weyl tensor]] and <math>Z</math> is the traceless Ricci tensor.

===Even-dimensional hypersurfaces===
For a compact, even-dimensional [[hypersurface]] <math> M </math> in <math> \mathbb{R}^{n+1} </math> we get<ref>{{cite book|last1=Guillemin |first1=V. |last2=Pollack |first2=A.| title=Differential topology | location=New York, NY |publisher=Prentice-Hall |year=1974 |page=196| isbn=978-0-13-212605-2}}</ref>

:<math>\int_M K\,dV = \frac{1}{2}\gamma_n\,\chi(M) </math>
where <math> dV </math> is the [[volume element]] of the hypersurface, <math>K</math> is the [[Jacobian matrix and determinant|Jacobian determinant]] of the [[Gauss map#Generalizations|Gauss map]], and <math>\gamma_n</math> is the [[N-sphere#Volume_and_surface_area|surface area of the unit n-sphere]].

=== Gauss–Bonnet theorem ===
{{Main|Gauss–Bonnet theorem}}
The [[Gauss–Bonnet theorem]] is a special case when <math> M </math> is a 2-dimensional manifold. It arises as the special case where the topological index is defined in terms of [[Betti number]]s and the analytical index is defined in terms of the Gauss–Bonnet integrand.

As with the two-dimensional Gauss–Bonnet theorem, there are generalizations when <math> M </math> is a [[manifold|manifold with boundary]].

==Further generalizations==

=== Atiyah–Singer ===
{{Main|Atiyah–Singer index theorem}}
A far-reaching generalization of the Gauss–Bonnet theorem is the [[Atiyah–Singer index theorem|Atiyah–Singer Index Theorem]].<ref name=":1" />

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 its [[adjoint operator]]. Then the '''analytical index''' is defined as

: <math>\dim(\ker(D))-\dim(\ker(D^*))</math>

By ellipticity this is always finite. The index theorem says that this is constant as the elliptic operator is varied smoothly. It is equal to a '''topological index''', which can be expressed in terms of [[characteristic class]]es like the [[Euler class]].

The Chern–Gauss–Bonnet theorem is derived by considering the [[Dirac operator]]

: <math>D = d + d^*</math>

=== Odd dimensions ===
The Chern formula is only defined for even dimensions because the [[Euler characteristic]] vanishes for odd dimensions. There is some research being done on 'twisting' the index theorem in [[K-theory]] to give non-trivial results for odd dimensions.<ref>{{Cite web |url=https://math.stackexchange.com/q/163287 |title=Why does the Gauss-Bonnet theorem apply only to even number of dimensons? |website=Mathematics Stack Exchange |date=June 26, 2012 |access-date=2019-05-08 }}</ref><ref>{{Cite arXiv <!-- |url=https://www.maths.ed.ac.uk/~v1ranick/papers/li4.pdf --> |title=The Gauss–Bonnet–Chern Theorem on Riemannian Manifolds|last=Li|first=Yin|year=2011|class=math.DG|eprint=1111.4972}}</ref>

There is also a version of Chern's formula for [[orbifold]]s.<ref>{{Cite web |url=https://mathoverflow.net/q/53302 |title=Is there a Chern-Gauss-Bonnet theorem for orbifolds? |website=MathOverflow |date=June 26, 2011 |access-date=2019-05-08}}</ref>

== History ==
[[Shiing-Shen Chern]] published his proof of the theorem in 1944 while at the [[Institute for Advanced Study]]. This was historically the first time that the formula was proven without assuming the manifold to be embedded in a Euclidean space, which is what it means by "intrinsic". The special case for a [[hypersurface]] (an (n-1)-dimensional submanifold in an n-dimensional Euclidean space) was proved by [[Heinz Hopf|H. Hopf]] in which the integrand is the Gauss–Kronecker curvature (the product of all principal curvatures at a point of the hypersurface). This was generalized independently by Allendoerfer in 1939 and Fenchel in 1940 to a Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz–Killing curvature (the average of the Gauss–Kronecker curvature along each unit normal vector over the unit sphere in the normal space; for an even dimensional submanifold, this is an invariant only depending on the Riemann metric of the submanifold). Their result would be valid for the general case if the [[Nash embedding theorem]] can be assumed. However, this theorem was not available then, as John Nash published his famous embedding theorem for Riemannian manifolds in 1956. In 1943 Allendoerfer and Weil published their proof for the general case, in which they first used an approximation theorem of H. Whitney to reduce the case to analytic Riemannian manifolds, then they embedded "small" neighborhoods of the manifold isometrically into a Euclidean space with the help of the Cartan–Janet local embedding theorem, so that they can patch these embedded neighborhoods together and apply the above theorem of Allendoerfer and Fenchel to establish the global result. This is, of course, unsatisfactory for the reason that the theorem only involves intrinsic invariants of the manifold, then the validity of the theorem should not rely on its embedding into a Euclidean space. Weil met Chern in Princeton after Chern arrived in August 1943. He told Chern that he believed there should be an intrinsic proof, which Chern was able to obtain within two weeks. The result is Chern's classic paper "A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds" published in the Annals of Mathematics the next year. The earlier work of Allendoerfer, Fenchel, Allendoerfer and Weil were cited by Chern in this paper. The work of Allendoerfer and Weil was also cited by Chern in his second paper related to the same topic.<ref name=":0" />

==See also==
{{Div col|colwidth=20em}}
*[[Chern–Weil homomorphism]]
*[[Chern class]]
*[[Chern–Simons form]]
*[[Chern–Simons theory]]
*[[Chern's conjecture (affine geometry)]]
*[[Pontryagin number]]
*[[Pontryagin class]]
*[[De Rham cohomology]]
*[[Berry's phase]]
*[[Atiyah–Singer index theorem]]
*[[Riemann–Roch theorem]]
{{Div col end}}

== References ==
{{Reflist}}

{{DEFAULTSORT:Generalized Gauss-Bonnet theorem}}
[[Category:Theorems in differential geometry]]