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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (4)

Label	Occurrences
Gauss–Bonnet theorem	7
Chern–Gauss–Bonnet theorem	2
Euler characteristic formula V−E+F=2	1
Gauss–Bonnet theorem (early form) canonical	1

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 ⓘ surface form: Disquisitiones generales circa superficies curvas
expresses	link between integral curvature and Euler characteristic for surfaces ⓘ
field	Riemannian manifolds ⓘ surface form: Riemannian geometry differential geometry ⓘ global differential geometry ⓘ
hasGeneralization	Chern–Weil theory ⓘ Chern–Weil theory ⓘ surface form: higher-dimensional Gauss–Bonnet formulas
historicalFormOf	Gauss–Bonnet theorem (early form) self-linksurface differs ⓘ surface form: Gauss–Bonnet theorem
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	Chern–Weil theory ⓘ surface form: modern Gauss–Bonnet theorem
relatedArea	algebraic topology ⓘ geometric analysis ⓘ
relatedTo	Chern–Weil theory ⓘ surface form: Chern–Gauss–Bonnet theorem Theorema Egregium ⓘ
relatesConcept	Euler’s polyhedron formula ⓘ surface form: Euler characteristic 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

The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.

Instruction

You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10.

# Requirements
- If you don't know the subject at all, return an empty list.
- If the subject is not a named entity, return an empty list.
- Include at least one triple where predicate is "instanceOf".
- Do not get too wordy.
- Separate several objects into multiple triples with one object.

Input

Subject: Gauss–Bonnet theorem (early form)
Description of subject: 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.