Theorema Egregium

E29359

mathematical theorem result in differential geometry

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (2)

Label	Occurrences
Theorema Egregium canonical	5
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	Theorema Egregium ⓘ surface form: Gauss’s 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 ⓘ surface form: Gaussian curvature is invariant under local isometries of surfaces
publicationYear	1828 ⓘ
publishedIn	Disquisitiones Generales Circa Superficies Curvas ⓘ
relatedTo	Gauss map ⓘ Gauss–Bonnet theorem (early form) ⓘ surface form: Gauss–Bonnet theorem 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

Referenced by (7)

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

Carl Friedrich Gauss → notableWork → Theorema Egregium ⓘ

Theorema Egregium → alsoKnownAs → Theorema Egregium ⓘ

this entity surface form: Gauss’s Theorema Egregium

Gauss–Bonnet theorem (early form) → relatedTo → Theorema Egregium ⓘ

Gaussian curvature → relatedTo → Theorema Egregium ⓘ

Disquisitiones Generales Circa Superficies Curvas → introducedConcept → Theorema Egregium ⓘ

Gauss’s remarkable theorem → hasAlternativeName → Theorema Egregium ⓘ