Editing Chern–Gauss–Bonnet theorem (section)

==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>