Gauss’s remarkable theorem

E157378

geometric theorem result in differential geometry theorem

Gauss’s remarkable theorem is a fundamental result in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.

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All labels observed (1)

Label	Occurrences
Gauss’s remarkable theorem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T1382045 — 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’s remarkable theorem
Context triple: [Theorema Egregium, alsoKnownAs, Gauss’s remarkable theorem]

A. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
B. Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
C. Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
D. Treatise on Demonstration of Problems of Algebra
Treatise on Demonstration of Problems of Algebra is a seminal mathematical work by Omar Khayyam in which he systematically analyzes and geometrically solves cubic equations.
E. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
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’s remarkable theorem
Target entity description: Gauss’s remarkable theorem is a fundamental result in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.

A. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
B. Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
C. Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
D. Treatise on Demonstration of Problems of Algebra
Treatise on Demonstration of Problems of Algebra is a seminal mathematical work by Omar Khayyam in which he systematically analyzes and geometrically solves cubic equations.
E. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	geometric theorem ⓘ result in differential geometry ⓘ theorem ⓘ
clarifies	difference between intrinsic and extrinsic properties of surfaces ⓘ
coreClaim	Gaussian curvature is independent of the embedding of the surface in Euclidean space ⓘ Gaussian curvature is preserved under local isometries of surfaces ⓘ Gaussian curvature of a surface is an intrinsic invariant ⓘ
dealsWith	Gaussian curvature ⓘ first fundamental form ⓘ intrinsic geometry ⓘ isometry of surfaces ⓘ second fundamental form ⓘ surfaces ⓘ
field	Riemannian geometry ⓘ differential geometry ⓘ
hasAlternativeName	Theorema Egregium ⓘ surface form: Gauss’s Theorema Egregium Theorema Egregium ⓘ
hasExampleApplication	proving that a cylinder is locally isometric to a plane ⓘ showing that a sphere is not locally isometric to a plane ⓘ understanding curvature of the Earth from geodesic measurements ⓘ
historicalPeriod	19th century ⓘ
implies	a plane cannot be bent isometrically into a sphere ⓘ bending a surface without stretching does not change its Gaussian curvature ⓘ curvature can be determined entirely from the metric on the surface ⓘ no isometric mapping exists between surfaces with different Gaussian curvature at corresponding points ⓘ
influenced	Bernhard Riemann ⓘ development of Riemannian geometry ⓘ general relativity ⓘ modern differential geometry ⓘ
introducedBy	Carl Friedrich Gauss ⓘ
languageOfOriginal	Latin ⓘ
mathematicalDomain	analysis on manifolds ⓘ geometry ⓘ
namedAfter	Carl Friedrich Gauss ⓘ
publicationYear	1827 ⓘ
publishedIn	Disquisitiones Generales Circa Superficies Curvas ⓘ surface form: Disquisitiones generales circa superficies curvas
relatesConcept	Christoffel symbols ⓘ Riemannian metric ⓘ extrinsic curvature ⓘ geodesic coordinates ⓘ intrinsic curvature ⓘ
status	fundamental theorem of surface theory ⓘ
typeOfCurvature	sectional curvature in dimension two ⓘ
usesConcept	determinant of the metric tensor ⓘ second derivatives of the metric ⓘ

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Full triples — surface form annotated when it differs from this entity's canonical label.

Theorema Egregium → alsoKnownAs → Gauss’s remarkable theorem ⓘ

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Statements (45)

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Referenced by (1)

How this entity was disambiguated

How these facts were elicited