Gauss–Codazzi equations
E653148
The Gauss–Codazzi equations are fundamental compatibility conditions in differential geometry that relate the intrinsic curvature of a surface to its extrinsic curvature as embedded in a higher-dimensional space.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gauss–Codazzi equations canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7290674 — 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–Codazzi equations Context triple: [Weingarten map, appearsIn, Gauss–Codazzi equations]
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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.
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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C.
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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D.
Cartan structure equations
Cartan structure equations are fundamental differential geometric relations that express curvature and torsion in terms of connection 1-forms on a manifold.
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E.
Weingarten map
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.
- 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–Codazzi equations Target entity description: The Gauss–Codazzi equations are fundamental compatibility conditions in differential geometry that relate the intrinsic curvature of a surface to its extrinsic curvature as embedded in a higher-dimensional space.
-
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.
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.
-
C.
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.
-
D.
Cartan structure equations
Cartan structure equations are fundamental differential geometric relations that express curvature and torsion in terms of connection 1-forms on a manifold.
-
E.
Weingarten map
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
geometric compatibility condition
ⓘ
result in differential geometry ⓘ system of equations ⓘ |
| appearsIn |
textbooks on Riemannian geometry
ⓘ
textbooks on general relativity ⓘ |
| appliesTo |
Riemannian submanifolds
NERFINISHED
ⓘ
embedded surfaces ⓘ |
| characterizes | possible embeddings with given first and second fundamental forms ⓘ |
| component |
Codazzi–Mainardi equations
NERFINISHED
ⓘ
Gauss equation NERFINISHED ⓘ |
| describes | relation between Riemann curvature tensor and second fundamental form ⓘ |
| domain |
Riemannian manifolds
ⓘ
pseudo-Riemannian manifolds ⓘ |
| ensures | consistency of intrinsic and extrinsic geometry ⓘ |
| expresses |
compatibility of first and second fundamental forms
ⓘ
integrability conditions for isometric immersion ⓘ |
| field |
Riemannian geometry
ⓘ
differential geometry ⓘ submanifold theory ⓘ |
| generalizes | Theorema Egregium NERFINISHED ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| holdsFor |
hypersurfaces
ⓘ
submanifolds of arbitrary codimension ⓘ |
| implies | constraints on curvature of embedded surfaces ⓘ |
| languageOfFormulation |
moving frames
ⓘ
tensor calculus ⓘ |
| namedAfter |
Carl Friedrich Gauss
NERFINISHED
ⓘ
Domenico Codazzi NERFINISHED ⓘ |
| relatedTo |
Gauss–Bonnet theorem
NERFINISHED
ⓘ
Weingarten equations NERFINISHED ⓘ fundamental theorem of surface theory NERFINISHED ⓘ |
| relates |
extrinsic curvature
ⓘ
intrinsic curvature ⓘ |
| requires |
Levi-Civita connection
NERFINISHED
ⓘ
Riemann curvature tensor NERFINISHED ⓘ second fundamental form ⓘ |
| type | tensorial equations ⓘ |
| usedIn |
ADM formalism
NERFINISHED
ⓘ
brane world models ⓘ computer graphics differential geometry ⓘ elasticity theory of shells ⓘ general relativity NERFINISHED ⓘ initial value formulation of Einstein field equations ⓘ shape analysis ⓘ theory of surfaces in Euclidean space ⓘ |
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–Codazzi equations Description of subject: The Gauss–Codazzi equations are fundamental compatibility conditions in differential geometry that relate the intrinsic curvature of a surface to its extrinsic curvature as embedded in a higher-dimensional space.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.