Gauss–Codazzi equations

E653148

geometric compatibility condition result in differential geometry system of equations

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.

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

All labels observed (3)

Label	Occurrences
Codazzi–Mainardi equations	1
Gauss equation	1
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]

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
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 ⓘ linked to: Riemannian manifolds embedded surfaces ⓘ
characterizes	possible embeddings with given first and second fundamental forms ⓘ
component	Codazzi–Mainardi equations ⓘ linked to: Gauss–Codazzi equations Gauss equation ⓘ linked to: Gauss–Codazzi equations
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 ⓘ
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 ⓘ Domenico Codazzi ⓘ
relatedTo	Gauss–Bonnet theorem ⓘ linked to: Gauss–Bonnet theorem (early form) Weingarten equations ⓘ linked to: Weingarten map fundamental theorem of surface theory ⓘ
relates	extrinsic curvature ⓘ intrinsic curvature ⓘ
requires	Levi-Civita connection ⓘ Riemann curvature tensor ⓘ second fundamental form ⓘ
type	tensorial equations ⓘ
usedIn	ADM formalism ⓘ brane world models ⓘ computer graphics differential geometry ⓘ elasticity theory of shells ⓘ general relativity ⓘ initial value formulation of Einstein field equations ⓘ shape analysis ⓘ theory of surfaces in Euclidean space ⓘ

How these facts were elicited

Referenced by (3)

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

Weingarten map → appearsIn → Gauss–Codazzi equations ⓘ

Gauss–Codazzi equations → component → Gauss equation ⓘ

linked to: Gauss–Codazzi equations

Gauss–Codazzi equations → component → Codazzi–Mainardi equations ⓘ