# Gauss–Codazzi 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. They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds.

In the classical differential geometry of surfaces, the Gauss–Codazzi–Mainardi equations consist of a pair of related equations. 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 equation is the basis for Gauss's theorema egregium.[1] The second equation, sometimes called the **Codazzi–Mainardi equation**, is a structural condition on the second derivatives of the Gauss map.
It was named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result,[2] although it was discovered earlier by Karl Mikhailovich Peterson.[3][4]
It incorporates the *extrinsic curvature* (or mean curvature) of the surface. 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.[5]

## Formal statement

Let be an *n*-dimensional embedded submanifold of a Riemannian manifold *P* of dimension . 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*:

The metric splits this short exact sequence, and so

Relative to this splitting, the Levi-Civita connection of *P* decomposes into tangential and normal components. For each and vector field *Y* on *M*,

Let

The **Gauss formula**[6] now asserts that is the Levi-Civita connection for *M*, and is a *symmetric* vector-valued form with values in the normal bundle. It is often referred to as the second fundamental form.

An immediate corollary is the **Gauss equation**. For ,

where is the Riemann curvature tensor of *P* and *R* is that of *M*.

The **Weingarten equation** is an analog of the Gauss formula for a connection in the normal bundle. Let and a normal vector field. Then decompose the ambient covariant derivative of along *X* into tangential and normal components:

Then

*Weingarten's equation*:*D*_{X}is a metric connection in the normal bundle.

There are thus a pair of connections: ∇, defined on the tangent bundle of *M*; and *D*, defined on the normal bundle of *M*. These combine to form a connection on any tensor product of copies of T*M* and T^{⊥}*M*. In particular, they defined the covariant derivative of :

The **Codazzi–Mainardi equation** is

Since every immersion is, in particular, a local embedding, the above formulas also hold for immersions.

## Gauss–Codazzi equations in classical differential geometry

### Statement of classical equations

In classical differential geometry of surfaces, the Codazzi–Mainardi equations are expressed via the second fundamental form (*L*, *M*, *N*):

### Derivation of classical equations

Consider a parametric surface in Euclidean 3-space,

where the three component functions depend smoothly on ordered pairs (*u*,*v*) in some open domain *U* in the *uv*-plane. Assume that this surface is **regular**, meaning that the vectors **r**_{u} and **r**_{v} are linearly independent. Complete this to a basis{**r**_{u},**r**_{v},**n**}, by selecting a unit vector **n** normal to the surface. It is possible to express the second partial derivatives of **r** using the Christoffel symbols and the second fundamental form.

Clairaut's theorem states that partial derivatives commute:

If we differentiate **r**_{uu} with respect to *v* and **r**_{uv} with respect to *u*, we get:

Now substitute the above expressions for the second derivatives and equate the coefficients of **n**:

Rearranging this equation gives the first Codazzi–Mainardi equation.

The second equation may be derived similarly.

## Mean curvature

Let *M* be a smooth *m*-dimensional manifold immersed in the (*m* + *k*)-dimensional smooth manifold *P*. Let be a local orthonormal frame of vector fields normal to *M*. Then we can write,

If, now, is a local orthonormal frame (of tangent vector fields) on the same open subset of *M*, then we can define the mean curvatures of the immersion by

In particular, if *M* is a hypersurface of *P*, i.e. , then there is only one mean curvature to speak of. The immersion is called minimal if all the are identically zero.

Observe that the mean curvature is a trace, or average, of the second fundamental form, for any given component. Sometimes mean curvature is defined by multiplying the sum on the right-hand side by .

We can now write the Gauss–Codazzi equations as

Contracting the components gives us

Observe that the tensor in parentheses is symmetric and nonnegative-definite in . Assuming that *M* is a hypersurface, this simplifies to

where and and . In that case, one more contraction yields,

where and are the respective scalar curvatures, and

If , the scalar curvature equation might be more complicated.

We can already use these equations to draw some conclusions. For example, any minimal immersion[7] into the round sphere must be of the form

where runs from 1 to and

is the Laplacian on *M*, and is a positive constant.

## See also

## Notes

- Gauss 1828.
- (Kline 1972, p. 885).
- Peterson (1853)
- Ivanov 2001.
- Bonnet 1867.
- Terminology from Spivak, Volume III.
- Takahashi 1966

## References

- Bonnet, Ossian (1867), "Memoire sur la theorie des surfaces applicables sur une surface donnee",
*Journal de l'École Polytechnique*,**25**: 31–151 - do Carmo, Manfredo Perdigao (1994),
*Riemannian Geometry*, Francis Flaherty - Codazzi, Delfino (1868–1869), "Sulle coordinate curvilinee d'una superficie dello spazio",
*Ann. Mat. Pura Appl.*,**2**: 101–19 - Gauss, Carl Friedrich (1828), "Disquisitiones Generales circa Superficies Curvas" [General Discussions about Curved Surfaces],
*Comm. Soc. Gott.*(in Latin),**6**("General Discussions about Curved Surfaces") - Ivanov, A.B. (2001) [1994], "Peterson–Codazzi equations", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Kline, Morris (1972),
*Mathematical Thought from Ancient to Modern Times*, Oxford University Press, ISBN 0-19-506137-3 - Mainardi, Gaspare (1856), "Su la teoria generale delle superficie",
*Giornale dell' Istituto Lombardo*,**9**: 385–404 - Peterson, Karl Mikhailovich (1853),
*Über die Biegung der Flächen*, Doctoral thesis, Dorpat University. - Takahashi, Tsunero (1966), "Minimal immersions of Riemannian manifolds",
*Journal of the Mathematical Society of Japan*