Editing Chern–Gauss–Bonnet theorem (section)

{{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.