About: Gauss–Codazzi equations

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In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.

Attributes	Values
rdfs:label	Gauss–Codazzi equations (en)
rdfs:comment	In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold. (en)
sameAs	Gauss–Codazzi equations
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Clarify dbt:ISBN dbt:Reflist dbt:Springer dbt:Curvature
Subject	Curvature (mathematics) Differential geometry of surfaces Riemannian geometry Surfaces
gold:hypernym	Equations
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Gauss–Codazzi_equations?oldid=1036061091&ns=0
Wikipage page ID	9633335 (xsd:integer)
page length (characters) of wiki page	13403 (xsd:nonNegativeInteger)
Wikipage revision ID	1036061091 (xsd:integer)
Link from a Wikipage to another Wikipage	Normal bundle Weingarten equations Basis (linear algebra) Tautology (logic) Carl Friedrich Gauss Curvature (mathematics) Differential geometry of surfaces Riemannian geometry Surfaces Differential geometry Metric connection Second fundamental form Christoffel symbols Delfino Codazzi Linear independence Pseudo-Riemannian manifold Pushforward (differential) Cokernel Covariant derivative Minimal surface Oxford University Press Parametric surface Darboux frame Gaspare Mainardi Gaussian curvature Immersion (mathematics) Karl Mikhailovich Peterson Symmetry of second derivatives Exact sequence Vector-valued differential form Euclidean space Laplace operator Levi-Civita connection Mean curvature Riemann curvature tensor Riemannian geometry Riemannian manifold Tangent bundle dbr:Journal_de_l'École_Polytechnique
has abstract	In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold. The equations were originally discovered in the context of surfaces in three-dimensional Euclidean space. In this context, the first equation, often called the Gauss equation (after its discoverer Carl Friedrich Gauss), says that the Gauss curvature of the surface, at any given point, is dictated by the derivatives of the Gauss map at that point, as encoded by the second fundamental form. The second equation, called the Codazzi equation or Codazzi-Mainardi equation, states that the covariant derivative of the second fundamental form is fully symmetric. It is named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, although it was discovered earlier by Karl Mikhailovich Peterson. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Gauss–Codazzi_equations
is rdfs:seeAlso of	Riemannian connection on a surface
is Wikipage redirect of	Gauss-Codazzi equations Gauss-Codazzi equations (relativity) Mainardi-Codazzi equation Mainardi-Codazzi equations Gauss-Codacci Gauss-Codazzi-Mainardi equations Gauss-Codazzi equation Gauss-Mainardi-Codazzi equations Gauss–Codazzi equation Gauss–Codazzi equations (relativity) Gauss–Codazzi–Mainardi equations Codazzi-Mainardi equation Codazzi equation
is Link from a Wikipage to another Wikipage of	List of things named after Carl Friedrich Gauss Bang-Yen Chen Biharmonic map Geodesic curvature Differential geometry of surfaces Christoffel symbols Delfino Codazzi Liouville's equation List of differential geometry topics List of scientific equations named after people Darboux frame Gaspare Mainardi Gauss-Codazzi equations Gauss-Codazzi equations (relativity) Index of physics articles (G) Karl Mikhailovich Peterson Mainardi-Codazzi equation Mainardi-Codazzi equations Ferdinand Minding Gauss-Codacci Gauss-Codazzi-Mainardi equations Gauss-Codazzi equation Gauss-Mainardi-Codazzi equations Gauss–Codazzi equation Gauss–Codazzi equations (relativity) Gauss–Codazzi–Mainardi equations Codazzi-Mainardi equation

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