Gauss–Bonnet theorem (early form)

E29918

mathematical theorem result in differential geometry theorem about curvature

The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.

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All labels observed (4)

Label	Occurrences
Gauss–Bonnet theorem	11
Chern–Gauss–Bonnet theorem	2
Euler characteristic formula V−E+F=2	1
Gauss–Bonnet theorem (early form) canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T228926 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Gauss–Bonnet theorem (early form)
Context triple: [Carl Friedrich Gauss, notableWork, Gauss–Bonnet theorem (early form)]

A. Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
B. Gaussian curvature
Gaussian curvature is a fundamental concept in differential geometry that measures how a surface bends at a point by combining its principal curvatures into a single intrinsic quantity.
C. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
D. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
E. Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Gauss–Bonnet theorem (early form)
Target entity description: The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.

A. Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
B. Gaussian curvature
Gaussian curvature is a fundamental concept in differential geometry that measures how a surface bends at a point by combining its principal curvatures into a single intrinsic quantity.
C. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
D. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
E. Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
F. None of above. chosen

Statements (43)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in differential geometry ⓘ theorem about curvature ⓘ
appliesTo	compact two-dimensional surfaces ⓘ smooth surfaces ⓘ
concerns	integral of curvature over a closed surface ⓘ topological invariants of surfaces ⓘ
coreIdea	integral of Gaussian curvature over a surface is determined by topological invariants ⓘ
developedBy	Carl Friedrich Gauss ⓘ
documentedIn	Disquisitiones generales circa superficies curvas ⓘ linked to: Disquisitiones Generales Circa Superficies Curvas
expresses	link between integral curvature and Euler characteristic for surfaces ⓘ
field	Riemannian geometry ⓘ linked to: Riemannian manifolds differential geometry ⓘ global differential geometry ⓘ
hasGeneralization	Chern–Weil theory ⓘ higher-dimensional Gauss–Bonnet formulas ⓘ linked to: Chern–Weil theory
historicalFormOf	Gauss–Bonnet theorem ⓘ linked to: Gauss–Bonnet theorem (early form)
importance	fundamental in differential geometry ⓘ
influenced	development of global differential geometry ⓘ topological methods in geometry ⓘ
involvesConcept	angle defect ⓘ geodesic triangles ⓘ intrinsic curvature ⓘ
involvesOperation	surface integral of curvature ⓘ
languageOfOriginalWork	Latin ⓘ
mathematicalSubjectClassification	53C20 ⓘ
namedAfter	Carl Friedrich Gauss ⓘ
predecessorOf	modern Gauss–Bonnet theorem ⓘ linked to: Chern–Weil theory
relatedArea	algebraic topology ⓘ geometric analysis ⓘ
relatedTo	Chern–Gauss–Bonnet theorem ⓘ linked to: Chern–Weil theory Theorema Egregium ⓘ
relatesConcept	Euler characteristic ⓘ linked to: Euler’s polyhedron formula Gaussian curvature ⓘ topology of surfaces ⓘ total curvature ⓘ
shows	curvature can be determined intrinsically ⓘ
status	proven theorem ⓘ
timePeriod	early 19th century ⓘ
topic	relationship between geometry and topology ⓘ
typeOfResult	global theorem ⓘ
usedIn	study of geodesic polygons ⓘ theory of polyhedral surfaces ⓘ

How these facts were elicited