Weingarten map
E164383
The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Weingarten map canonical | 3 |
| Weingarten equations | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1427934 — 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: Weingarten map Context triple: [Gauss map, relatedConcept, Weingarten map]
-
A.
Gauss map
The Gauss map is a differential geometry concept that assigns to each point on a surface the corresponding point on the unit sphere determined by the surface’s normal vector at that point.
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B.
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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C.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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D.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
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E.
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.
- 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: Weingarten map Target entity description: The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
-
A.
Gauss map
The Gauss map is a differential geometry concept that assigns to each point on a surface the corresponding point on the unit sphere determined by the surface’s normal vector at that point.
-
B.
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.
-
C.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
D.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
differential geometric operator
ⓘ
linear map ⓘ shape operator ⓘ |
| actsOn | tangent space of a surface ⓘ |
| alsoKnownAs | shape operator ⓘ |
| appearsIn |
Gauss–Codazzi equations
ⓘ
theory of hypersurfaces in space forms ⓘ |
| appliesTo |
oriented hypersurfaces
ⓘ
oriented surfaces ⓘ |
| category | linear operators in differential geometry ⓘ |
| codomain | tangent space of a surface at a point ⓘ |
| context | extrinsic geometry of submanifolds ⓘ |
| definedAs | S(X) = -∇_X n where n is the unit normal ⓘ |
| definedFor |
hypersurfaces in Riemannian manifolds
ⓘ
regular surfaces in Euclidean space ⓘ |
| dependsOn | choice of unit normal field ⓘ |
| determinantRelation | determinant equals Gauss curvature for surfaces in R^3 ⓘ |
| domain | tangent space of a surface at a point ⓘ |
| eigenvalues | principal curvatures ⓘ |
| eigenvectors | principal directions ⓘ |
| encodes |
extrinsic curvature of a surface
ⓘ
variation of the unit normal vector field ⓘ |
| fieldOfStudy |
Riemannian geometry
ⓘ
differential geometry ⓘ surface theory ⓘ |
| generalizationOf | curvature of plane curves to higher dimensions ⓘ |
| introducedBy | Julius Weingarten ⓘ |
| invariantUnder | isometries of the ambient Euclidean space ⓘ |
| mathematicalDefinition | negative of the differential of the Gauss map ⓘ |
| namedAfter | Julius Weingarten ⓘ |
| property |
diagonalizable over the reals for regular surfaces
ⓘ
self-adjoint with respect to the induced metric ⓘ |
| relatedConcept |
Gauss map
ⓘ
normal bundle ⓘ second fundamental form matrix ⓘ |
| relatedTo |
Gauss curvature
ⓘ
mean curvature ⓘ second fundamental form ⓘ |
| requires | Levi-Civita connection of the ambient space ⓘ |
| symbol |
A
ⓘ
S ⓘ |
| traceRelation | trace equals 2 times mean curvature for surfaces in R^3 ⓘ |
| usedFor |
classifying points as elliptic hyperbolic or parabolic
ⓘ
computing curvature invariants ⓘ studying stability of minimal surfaces ⓘ |
| yearOfIntroduction | 19th century ⓘ |
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: Weingarten map Description of subject: The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Weingarten map
linked to: Weingarten map