Gauss map
E30376
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gauss map canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T228957 — 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 map Context triple: [Carl Friedrich Gauss, hasConceptNamedAfter, Gauss map]
-
A.
Gaussian curvature
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.
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B.
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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C.
Gauss–Bonnet theorem (early form)
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.
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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.
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 map Target entity description: 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.
-
A.
Gaussian curvature
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.
-
B.
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.
-
C.
Gauss–Bonnet theorem (early form)
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.
-
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.
-
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 |
differential geometry concept
ⓘ
map between manifolds ⓘ |
| ambientSpace | Euclidean 3-space ⓘ |
| appliesTo |
immersed surfaces
ⓘ
smooth surfaces ⓘ |
| canBe | local diffeomorphism at non-umbilic points ⓘ |
| category |
geometric mapping
ⓘ
surface invariant ⓘ |
| codomain |
2-sphere
ⓘ
unit sphere ⓘ |
| criticalPoints | umbilic points of the surface ⓘ |
| definition | map that assigns to each point on a surface the endpoint of the unit normal vector on the unit sphere ⓘ |
| dependsOn | choice of orientation of the surface ⓘ |
| domain | regular surface in Euclidean 3-space ⓘ |
| field |
Riemannian manifolds
ⓘ
surface form:
Riemannian geometry
classical surface theory ⓘ differential geometry ⓘ |
| generalization |
Gauss map of hypersurfaces in higher-dimensional Euclidean spaces
ⓘ
normal map in Riemannian geometry ⓘ |
| hasVariant | spherical image of a surface ⓘ |
| input | point on a surface ⓘ |
| introducedBy | Carl Friedrich Gauss ⓘ |
| inverseImage | set of points on surface with same normal direction ⓘ |
| isSmooth | yes ⓘ |
| mapsTo | unit normal vector direction ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| output | point on the unit sphere ⓘ |
| property |
Jacobian determinant equals Gaussian curvature up to sign
ⓘ
differential equals negative of shape operator ⓘ |
| relatedConcept |
Gaussian curvature
ⓘ
Weingarten map ⓘ second fundamental form ⓘ shape operator ⓘ |
| requires |
choice of unit normal field
ⓘ
oriented surface ⓘ |
| studiedIn | classical differential geometry of curves and surfaces ⓘ |
| symbol |
G
ⓘ
N ⓘ |
| theorem |
Gauss–Bonnet theorem uses integral of Gaussian curvature derived from Gauss map
ⓘ
area of image under Gauss map relates to total curvature ⓘ |
| usedFor |
definition of Gaussian curvature
ⓘ
definition of shape operator ⓘ measuring how a surface bends in space ⓘ studying extrinsic geometry of 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: Gauss map Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.