Gauss’s remarkable theorem
E157378
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.
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.
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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.
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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.
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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.
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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 |
Gauss’s Theorema Egregium
ⓘ
linked to:
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 ⓘ |
| 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 ⓘ |
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’s remarkable theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.