Gauss–Codazzi equations

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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: see Gauss-Codazzi equations (relativity).

In the classical geometry of surfaces, the Gauss–Codazzi–Mainardi equations consist of a pair of related equations. The first equation, sometimes called the Gauss equation 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.^[2] 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.^[3]

Contents 1 Formal statement 2 Statement of classical equations 3 Derivation of classical equations 4 History 5 See also 6 External links 7 Notes 8 References

[edit] Formal statement

Let i : M ⊂ P be an n-dimensional embedded submanifold of a Riemannian manifold P of dimension n+p. 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 X ∈ TM and vector field Y on M,

Let

Gauss' formula^[4] now asserts that ∇_X is the Levi-Civita connection for M, and α is a symmetric vector-valued form with values in the normal bundle.

An immediate corollary is the Gauss equation. For X, Y, Z, W ∈ TM,

where R′ 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 X ∈ TM and ξ a normal vector field. Then decompose the ambient covariant derivative of ξ along X into tangential and normal components:

Then

1. Weingarten's equation:
2. 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 TM and T^⊥M. In particular, they defined the covariant derivative of α:

The Codazzi-Mainardi equation is

[edit] Statement of classical equations

In classical differential geometry of surfaces, the Codazzi-Mainardi equations are expressed via the second fundamental form:

[edit] Derivation of classical equations

The second derivatives of a parametric surface may be expressed in the basis {X_u,X_v,N} with the Christoffel symbols and the second fundamental form.

Clairaut's theorem states that partial derivatives commute:

If we differentiate X_uu with respect to v and X_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.

[edit] History

First were discovered by Peterson^[5], then Mainardi^[6] and Delfino Codazzi (1867) ^[7].

[edit] See also

• Darboux frame

[edit] External links

• Peterson-Mainardi-Codazzi Equations -- from Wolfram MathWorld
• Peterson-Codazzi Equations

[edit] Notes

1. ^ Gauss (1828).
2. ^ Named for Gaspare Mainardi (1856) and Delfino Codazzi (1868-1869), who independently derived the result. Cf. Kline (1972), p. 885.
3. ^ Bonnet (1867).
4. ^ Terminology from Spivak, Volume III.
5. ^ Peterson, K. M. "Über die Biegung der Flächen." Dorpat. Kandidatenschrift. 1853.
6. ^ Mainardi, G. "Sulle coordinate curvilinee d'una superfice dello spazio." Giornale del R. Istituto Lombardo 9, 385-398, 1856.
7. ^ Codazzi, D. "Sulle coordinate curvilinee d'una superficie dello spazio." Ann. math. pura applicata 2, 101-19, 1868-1869.

[edit] References

• Bonnet, Ossian (1867). "Memoire sur la theorie des surfaces applicables sur une surface donnee". Jour. de l'Ecole Poly. 25: 31-151. 
• do Carmo, Manfredo Perdigao (1994). Riemannian Geometry. 
• Gauss, Carl Friedrich (1828). "Disquitiones Generales circa Superficies Curvas" (in Latin). Comm. Soc. Gott. 6.  ("General Discussions about Curved Surfaces")
• Codazzi, Delfino (1868-1869). "Sulle coordinate curvilinee d'una superficie dello spazio". Ann. math. pura applicata 2: 101-19. 
• 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. 

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