Gaussian curvature
E29546
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gaussian curvature canonical | 2 |
| Gaussian curvature is invariant under local isometries of surfaces | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T228924 — 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: Gaussian curvature Context triple: [Carl Friedrich Gauss, notableWork, Gaussian curvature]
-
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.
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B.
Ricci curvature tensor
The Ricci curvature tensor is a geometric object in differential geometry that measures how volumes in a curved space-time deviate from those in flat space, playing a central role in general relativity.
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C.
Riemann curvature tensor
The Riemann curvature tensor is a fundamental geometric object in differential geometry that measures how much a Riemannian manifold deviates from being flat by encoding the intrinsic curvature of the space.
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D.
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.
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E.
Kretschmann scalar
The Kretschmann scalar is a curvature invariant in general relativity that combines components of the Riemann tensor into a single scalar quantity used to characterize the intensity of spacetime curvature, especially near singularities.
- 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: Gaussian curvature Target entity description: 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.
-
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.
Ricci curvature tensor
The Ricci curvature tensor is a geometric object in differential geometry that measures how volumes in a curved space-time deviate from those in flat space, playing a central role in general relativity.
-
C.
Riemann curvature tensor
The Riemann curvature tensor is a fundamental geometric object in differential geometry that measures how much a Riemannian manifold deviates from being flat by encoding the intrinsic curvature of the space.
-
D.
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.
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E.
Kretschmann scalar
The Kretschmann scalar is a curvature invariant in general relativity that combines components of the Riemann tensor into a single scalar quantity used to characterize the intensity of spacetime curvature, especially near singularities.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
curvature invariant
ⓘ
differential geometry concept ⓘ scalar field on a surface ⓘ |
| appearsIn |
Riemannian surface classification
ⓘ
surface theory ⓘ |
| canBe | constant on some surfaces ⓘ |
| characterizes | local shape of a surface ⓘ |
| codomain | real numbers ⓘ |
| definedAs | product of principal curvatures ⓘ |
| dependsOn |
first fundamental form
ⓘ
second fundamental form ⓘ |
| describes | intrinsic curvature of a surface ⓘ |
| domain | points of a regular surface ⓘ |
| exampleConstantCurvatureSurface |
Euclidean space
ⓘ
surface form:
Euclidean plane has K = 0
hyperbolic plane has K < 0 ⓘ sphere of radius R has K = 1/R^2 ⓘ |
| exampleSurfaceWithNegativeCurvature |
hyperbolic plane
ⓘ
pseudosphere ⓘ |
| exampleSurfaceWithPositiveCurvature | sphere ⓘ |
| exampleSurfaceWithZeroCurvature |
cylinder
ⓘ
plane ⓘ |
| field |
Riemannian manifolds
ⓘ
surface form:
Riemannian geometry
differential geometry ⓘ |
| formulaInPrincipalDirections | K = k1 * k2 ⓘ |
| isExtrinsic | false ⓘ |
| isIntrinsic | true ⓘ |
| mathematicalNature | local invariant of a 2-dimensional Riemannian manifold ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| property |
can be computed from the metric alone
ⓘ
invariant under local isometries of surfaces ⓘ |
| relatedTo |
Gauss–Bonnet theorem (early form)
ⓘ
surface form:
Gauss–Bonnet theorem
Theorema Egregium ⓘ mean curvature ⓘ principal curvatures ⓘ sectional curvature ⓘ |
| signInterpretation |
negative at hyperbolic points
ⓘ
positive at elliptic points ⓘ zero at parabolic points ⓘ |
| symbol | K ⓘ |
| unit | inverse square of length ⓘ |
| usedIn |
classification of points on a surface
ⓘ
general relativity analogues in 2D ⓘ geodesic analysis ⓘ global topology via Gauss–Bonnet theorem ⓘ |
| valueType |
negative curvature
ⓘ
positive curvature ⓘ zero curvature ⓘ |
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: Gaussian curvature Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gaussian curvature is invariant under local isometries of surfaces