Janet–Cartan theorem
E3651
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Janet–Cartan theorem canonical | 3 |
| Cartan–Janet system of partial differential equations | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T31661 — 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: Janet–Cartan theorem Context triple: [Nash embedding theorem, relatedTo, Janet–Cartan theorem]
-
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.
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B.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
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C.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
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D.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
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E.
Differential analyzer
The Differential Analyzer is an early analog mechanical computer designed to solve differential equations using interconnected rotating shafts and wheels.
- 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: Janet–Cartan theorem Target entity description: 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.
-
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.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
C.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
-
D.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
-
E.
Heavenly Gondola
Heavenly Gondola is a scenic aerial tramway at Heavenly Mountain Resort that transports visitors between South Lake Tahoe and the resort’s mountain slopes, offering panoramic views of the lake and surrounding Sierra Nevada.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in differential geometry ⓘ |
| appliesTo | real-analytic Riemannian manifolds ⓘ |
| assumes | real-analytic Riemannian metric ⓘ |
| assumption |
positive-definite Riemannian metric
ⓘ
real-analytic structure on the manifold ⓘ |
| category | embedding theorems in geometry ⓘ |
| codomain | Euclidean space R^N ⓘ |
| concerns |
Euclidean space
ⓘ
Riemannian manifolds ⓘ local isometric embeddings ⓘ |
| contrastWith | Nash embedding theorem which works for C^k metrics ⓘ |
| dimensionBound | N = n(3n+11)/2 for an n-dimensional manifold ⓘ |
| dimensionBoundType | local embedding dimension upper bound ⓘ |
| doesNotGenerallyApplyTo | smooth non-analytic Riemannian manifolds ⓘ |
| embeddingType | local isometric embedding ⓘ |
| ensures | local isometry between the manifold and its image in Euclidean space ⓘ |
| field |
Riemannian geometry
ⓘ
differential geometry ⓘ global analysis ⓘ |
| guarantees | existence of local isometric embeddings into Euclidean space ⓘ |
| historicalPeriod | early 20th century ⓘ |
| implies | local realization of analytic Riemannian metrics as induced metrics from Euclidean space ⓘ |
| influenced | later work on isometric embeddings ⓘ |
| language | originally formulated in French ⓘ |
| mathematicalSubjectClassification |
53C21
ⓘ
58J32 ⓘ |
| method | use of analytic partial differential equations ⓘ |
| metricPreserved |
Riemannian distance locally
ⓘ
inner products of tangent vectors ⓘ |
| namedAfter |
Maurice Janet
ⓘ
Élie Cartan ⓘ |
| originalContext | study of integrability conditions for systems of PDEs ⓘ |
| relatedTo |
Cartan–Janet system of partial differential equations
ⓘ
linked to:
Janet–Cartan theorem
Nash embedding theorems ⓘ
linked to:
Nash embedding theorem
isometric embedding problem ⓘ |
| requires | analyticity of the metric coefficients in local coordinates ⓘ |
| resultType | existence theorem ⓘ |
| scope | local embeddings rather than global embeddings ⓘ |
| states | any n-dimensional real-analytic Riemannian manifold can be locally isometrically embedded into some Euclidean space R^N ⓘ |
| strengthenedBy |
Nash C^1 isometric embedding theorem
ⓘ
linked to:
Nash embedding theorem
Nash C^k isometric embedding theorem ⓘ
linked to:
Nash embedding theorem
|
| topic | local geometry of Riemannian manifolds ⓘ |
| usedIn |
the study of local realizability of metrics
ⓘ
the theory of overdetermined PDE systems ⓘ |
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: Janet–Cartan theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Janet–Cartan theorem