Editing Gauss–Bonnet theorem (section)

{{Short description|Theorem in differential geometry}}
{{redirect-distinguish|Gauss–Bonnet|Gauss–Bonnet gravity}}
{{more citations needed|date=October 2020}}
[[File:Gauss-Bonnet theorem.svg|thumb|300px|An example of a complex region where Gauss–Bonnet theorem can apply. Shows the sign of geodesic curvature.]]
In the mathematical field of [[differential geometry]], the '''Gauss–Bonnet theorem''' (or '''Gauss–Bonnet formula''') is a fundamental formula which links the [[curvature]] of a [[Surface (topology)|surface]] to its underlying [[topology]].

In the simplest application, the case of a triangle [[Euclidean geometry|on a plane]], the [[Sum of angles of a triangle|sum of its angles]] is 180 degrees.<ref>{{Cite interview |last=Chern |first=Shiing-Shen |subject-link=Shiing-Shen Chern |interviewer=Allyn Jackson |title=Interview with Shiing-Shen Chern |url=https://www.ams.org/notices/199807/chern.pdf |access-date=2019-07-22 |date=March 4, 1998}}</ref> The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries.

The theorem is named after [[Carl Friedrich Gauss]], who developed a version but never published it, and [[Pierre Ossian Bonnet]], who published a special case in 1848.{{not verified in body|date=October 2020}}