Gauss–Codazzi equations

Gauss-Codazzi equationsGauss–Codazzi equationGauss–Codazzi equations (relativity)Codazzi–Mainardi equationsGauss equationGauss–Codazzi–MainardiGauss–Codazzi–Mainardi equations
In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds.wikipedia
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Differential geometry of surfaces

surfaceshape operatorsmooth surface
In the classical differential geometry of surfaces, the Gauss–Codazzi–Mainardi equations consist of a pair of related equations.
The coefficients of the first and second fundamental forms satisfy certain compatibility conditions known as the Gauss-Codazzi equations;

Karl Mikhailovich Peterson

Karl PetersonPeterson, Karl Mikhailovich
It was named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, although it was discovered earlier by Karl Mikhailovich Peterson.
Karl Mikhailovich Peterson (1828–1881) was a Russian mathematician, known by an earlier formulation of the Gauss–Codazzi equations.

Delfino Codazzi

Codazzi, Delfino
It was named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, although it was discovered earlier by Karl Mikhailovich Peterson.
He made some important contributions to the differential geometry of surfaces, such as the Codazzi–Mainardi equations.

Second fundamental form

extrinsic curvaturesecondshape tensor
The first equation, sometimes called the Gauss equation (named after Carl Friedrich Gauss), relates the intrinsic curvature (or Gauss curvature) of the surface to the derivatives of the Gauss map, via the second fundamental form.
:This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium.

Gaspare Mainardi

Mainardi, Gaspare
It was named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, although it was discovered earlier by Karl Mikhailovich Peterson.
He is remembered for the Gauss–Codazzi–Mainardi equations.

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
It is possible to express the second partial derivatives of r using the Christoffel symbols and the second fundamental form.

Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds.

Hypersurface

complex hypersurfaceprojective hypersurfacesurface
In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds.

Euclidean space

EuclideanspaceEuclidean vector space
In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds.

Submanifold

immersed submanifoldembedded submanifoldembedded
In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds.

Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudo
They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds.

Carl Friedrich Gauss

GaussCarl GaussCarl Friedrich Gauß
The first equation, sometimes called the Gauss equation (named after Carl Friedrich Gauss), relates the intrinsic curvature (or Gauss curvature) of the surface to the derivatives of the Gauss map, via the second fundamental form.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
The first equation, sometimes called the Gauss equation (named after Carl Friedrich Gauss), relates the intrinsic curvature (or Gauss curvature) of the surface to the derivatives of the Gauss map, via the second fundamental form.

Gauss map

The first equation, sometimes called the Gauss equation (named after Carl Friedrich Gauss), relates the intrinsic curvature (or Gauss curvature) of the surface to the derivatives of the Gauss map, via the second fundamental form.

Theorema Egregium

This equation is the basis for Gauss's theorema egregium.

Mean curvature

average curvaturemeanmean radius of curvature
It incorporates the extrinsic curvature (or mean curvature) of the surface.

Rigid transformation

Euclidean transformationrigid motionEuclidean isometries
The equations show that the components of the second fundamental form and its derivatives along the surface completely classify the surface up to a Euclidean transformation, a theorem of Ossian Bonnet.

Pierre Ossian Bonnet

BonnetOssian BonnetPierre Bonnet
The equations show that the components of the second fundamental form and its derivatives along the surface completely classify the surface up to a Euclidean transformation, a theorem of Ossian Bonnet.

Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
There is a natural inclusion of the tangent bundle of M into that of P by the pushforward, and the cokernel is the normal bundle of M:

Pushforward (differential)

pushforwarddifferentialderivative
There is a natural inclusion of the tangent bundle of M into that of P by the pushforward, and the cokernel is the normal bundle of M:

Cokernel

cokernelscategory-theoretic cokernelco''kernel
There is a natural inclusion of the tangent bundle of M into that of P by the pushforward, and the cokernel is the normal bundle of M:

Normal bundle

conormal bundlenormalnormal sheaf
There is a natural inclusion of the tangent bundle of M into that of P by the pushforward, and the cokernel is the normal bundle of M:

Exact sequence

short exact sequencelong exact sequenceexact
:The metric splits this short exact sequence, and so

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
Relative to this splitting, the Levi-Civita connection \nabla' of P decomposes into tangential and normal components.