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.

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

Referenced by (1)

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

Weingarten map → appearsIn → Gauss–Codazzi equations ⓘ