Editing Chern–Gauss–Bonnet theorem (section)

== 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.).