Janet–Cartan theorem

E3651

mathematical theorem theorem in differential geometry

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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Observed surface forms (1)

Surface form	Occurrences
Cartan–Janet system of partial differential equations	1

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	Janet–Cartan theorem self-linksurface differs ⓘ surface form: Cartan–Janet system of partial differential equations Nash embedding theorem ⓘ surface form: Nash embedding theorems 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 embedding theorem ⓘ surface form: Nash C^1 isometric embedding theorem Nash embedding theorem ⓘ surface form: Nash C^k isometric embedding theorem
topic	local geometry of Riemannian manifolds ⓘ
usedIn	the study of local realizability of metrics ⓘ the theory of overdetermined PDE systems ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Maurice Janet → hasNotableTheoremNamedAfter → Janet–Cartan theorem ⓘ

Maurice Janet → notableWork → Janet–Cartan theorem ⓘ

Janet–Cartan theorem → relatedTo → Janet–Cartan theorem self-linksurface differs ⓘ

this entity surface form: Cartan–Janet system of partial differential equations

Nash embedding theorem → relatedTo → Janet–Cartan theorem ⓘ