Theorema Egregium
E29359
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Theorema Egregium canonical | 6 |
| Gauss’s Theorema Egregium | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T228918 — 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: Theorema Egregium Context triple: [Carl Friedrich Gauss, notableWork, Theorema Egregium]
-
A.
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.
-
B.
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.
-
C.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
D.
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.
-
E.
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.
- 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: Theorema Egregium Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in differential geometry ⓘ |
| alsoKnownAs |
Gauss’s Theorema Egregium
ⓘ
linked to:
Theorema Egregium
Gauss’s remarkable theorem ⓘ |
| appliesTo |
regular surfaces
ⓘ
smooth surfaces ⓘ |
| author | Carl Friedrich Gauss ⓘ |
| concerns |
curved surfaces in three-dimensional Euclidean space
ⓘ
intrinsic curvature ⓘ metric properties of surfaces ⓘ two-dimensional surfaces ⓘ |
| context | classical differential geometry of surfaces ⓘ |
| dimension | two-dimensional manifolds ⓘ |
| field |
Riemannian geometry
ⓘ
differential geometry ⓘ |
| historicalSignificance | first clear demonstration of intrinsic curvature of surfaces ⓘ |
| implies |
Gaussian curvature can be computed from the first fundamental form alone
ⓘ
local isometry preserves Gaussian curvature ⓘ |
| influenced |
concept of intrinsic curvature in general relativity
ⓘ
development of Riemannian geometry ⓘ |
| mainConcept |
Gaussian curvature
ⓘ
first fundamental form ⓘ intrinsic geometry ⓘ second fundamental form ⓘ |
| mathematicalSubjectClassification | 53A05 ⓘ |
| namedAfter | Latin phrase meaning remarkable theorem ⓘ |
| originalLanguage | Latin ⓘ |
| proves |
Gaussian curvature is invariant under local isometries of surfaces
ⓘ
linked to:
Gaussian curvature
|
| publicationYear | 1828 ⓘ |
| publishedIn | Disquisitiones Generales Circa Superficies Curvas ⓘ |
| relatedTo |
Gauss map
ⓘ
Gauss–Bonnet theorem ⓘ
linked to:
Gauss–Bonnet theorem (early form)
first fundamental form ⓘ second fundamental form ⓘ |
| shows |
curvature of a surface can be determined by measurements within the surface
ⓘ
extrinsic curvature is not needed to determine Gaussian curvature ⓘ |
| statedBy | Carl Friedrich Gauss ⓘ |
| statesThat |
Gaussian curvature is independent of the embedding of the surface in Euclidean space
ⓘ
Gaussian curvature of a surface is an intrinsic invariant ⓘ |
| typeOfInvariance | intrinsic invariance ⓘ |
| usesConcept |
Christoffel symbols
ⓘ
coefficients of the first fundamental form ⓘ metric tensor on a surface ⓘ |
| yearProved | 1827 ⓘ |
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: Theorema Egregium Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Theorema Egregium
linked to: Theorema Egregium