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.

Aliases (3)

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 →
expresses	link between integral curvature and Euler characteristic for surfaces →
field	Riemannian geometry → differential geometry → global differential geometry →
hasGeneralization	Chern–Weil theory → higher-dimensional Gauss–Bonnet formulas →
historicalFormOf	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	modern Gauss–Bonnet theorem →
relatedArea	algebraic topology → geometric analysis →
relatedTo	Chern–Gauss–Bonnet theorem → Theorema Egregium →
relatesConcept	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 →

Referenced by (8)

Subject (surface form when different)	Predicate
Carl Friedrich Gauss → Leonhard Euler ("Euler characteristic formula V−E+F=2") →	notableWork
Gaussian curvature ("Gauss–Bonnet theorem") → Theorema Egregium ("Gauss–Bonnet theorem") →	relatedTo
Atiyah–Singer index theorem ("Gauss–Bonnet theorem") →	generalizes
Carl Friedrich Gauss ("Gauss–Bonnet theorem") →	hasConceptNamedAfter
Gauss–Bonnet theorem (early form) ("Gauss–Bonnet theorem") →	historicalFormOf
Shiing-Shen Chern ("Chern–Gauss–Bonnet theorem") →	knownFor