# 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**

### Riemannian manifold

**Riemannian metricRiemannianRiemannian 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

### 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**

### 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.