Gauss-Weingarten equations

The Gauß-Weingarten equations (after Carl Friedrich Gauß and Julius Weingarten ) are a system of partial differential equations from differential geometry . They convey a relationship between the tangential vectors , the unit normal of a regular surface and the coefficients of the matrix of the first or the second fundamental form with regard to a (local) parameterization of this surface. ${\displaystyle X_{1},X_{2}}$ ${\displaystyle N={\tfrac {X_{1}\times X_{2}}{|X_{1}\times X_{2}|}}}$

Equations

The equations are (i, j, k = 1,2):

${\displaystyle X_{ij}=\Gamma _{ij}^{k}X_{k}+l_{ij}N,\qquad N_{i}=-l_{ij}g^{jk}X_{k}.}$

The vectors stand for ${\displaystyle X_{1},X_{2}}$

${\displaystyle X_{1}=X_{u}(u,v)={\tfrac {\partial X}{\partial u}}(u,v)\quad {\text{und}}\quad X_{2}=X_{v}(u,v)={\tfrac {\partial X}{\partial v}}(u,v)}$

the first partial derivatives according to the parameters or the area and correspondingly (i, j = 1,2) for the second derivatives. Correspondingly, (i = 1,2) are the derivatives of the normal vector. ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle X_{ij}}$ ${\displaystyle N_{i}}$

If we take into account that the vectors are linearly independent for a surface that is regular in differential geometry , then we can represent the first derivatives of this tripod as a linear combination of the basis vectors. The Gauss-Weingarten equations then provide a determination of the coefficients. ${\displaystyle X_{1},X_{2},N}$

The are the Christoffel symbols of the coefficients of the matrix of the first fundamental form with the coefficients of the inverse matrix and the coefficients of the matrix of the second fundamental form (often , , written). Those are the coefficients of the vineyard map . ${\displaystyle \Gamma _{ij}^{k}}$ ${\displaystyle g_{ij}}$ ${\displaystyle g^{kl}}$ ${\displaystyle l_{ij}}$ ${\displaystyle l_{11}=L}$ ${\displaystyle l_{12}=l_{21}=M}$ ${\displaystyle l_{22}=N}$ ${\displaystyle l_{ij}g^{jk}}$

Originally no Christoffels symbols were used in the formulas, but the coefficients of the equation were expressed by the coefficients of the first fundamental form of the surface , and . With the discriminant of the fundamental form and the first derivatives etc. the following relationships apply: ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle g=EG-F^{2}}$ ${\displaystyle E_{u}}$

${\displaystyle \Gamma _{11}^{1}={\frac {1}{2g}}(GE_{u}-2FF_{u}+FE_{v})}$

${\displaystyle \Gamma _{12}^{1}=\Gamma _{21}^{1}={\frac {1}{2g}}(GE_{v}-FG_{u})}$

${\displaystyle \Gamma _{22}^{1}={\frac {1}{2g}}(-FG_{v}+2GF_{v}-GG_{u})}$

${\displaystyle \Gamma _{11}^{2}={\frac {1}{2g}}(-FE_{u}+2EF_{u}-EE_{v})}$

${\displaystyle \Gamma _{12}^{2}=\Gamma _{21}^{2}={\frac {1}{2g}}(EG_{u}-FE_{v})}$

${\displaystyle \Gamma _{22}^{2}={\frac {1}{2g}}(EG_{v}-2FF_{v}+FG_{u})}$

The coefficients of the vineyard map are written accordingly: ${\displaystyle -l_{ij}g^{jk}}$

• i = 1, k = 1: ${\displaystyle {\frac {(MF-LG)}{g}}}$
• i = 1, k = 2: ${\displaystyle {\frac {(LF-ME)}{g}}}$
• i = 2, k = 1: ${\displaystyle {\frac {(NF-MG)}{g}}}$
• i = 2, k = 2: ${\displaystyle {\frac {(MF-NE)}{g}}}$

Integration conditions

The question arises to what extent a differential-geometrically regular surface is (uniquely) determined by specifying the first and second fundamental form. If one computes mixed second derivatives of the tripod, one finds that the coefficients of the first and second fundamental form cannot be chosen completely independently of one another. The necessary integration conditions apply in the form of the Codazzi-Mainardi equations and Brioschi's formula . It is found that the necessary conditions are also sufficient. The fundamental theorem of area theory applies:

The coefficients of the matrix of the first and second Fundemantal form satisfy the Codazzi-Mainardi equations and Brioschi's formula. Then there is a uniquely determined surface, except for translations and rotations, which has the prescribed first and second fundamental form.

The Gauß-Weingarten equations represent the generalization of Frenet's formulas for surfaces in three-dimensional space. The part of the formulas with the derivation of the normal vector is also called derivation formulas by Weingarten (1861).

Generalizations

The original version of the Gauss-Weingarten equations only applies to two-dimensional differentiable manifolds in three-dimensional space. One can write down the equations without further problems for general differentiable manifolds with codimension 1, that is for hypersurfaces . To do this, a base of the tangential bundle is supplemented point by point by a unit normal vector and thus a base of the n-dimensional space is obtained. With the analogous method, the Gauss-Weingarten equations are then represented for these manifolds.

There are also suitable generalizations in higher code dimensions . To do this, a basis of a tangential bundle is again supplemented by corresponding unit normal vectors . However, these must be chosen in such a way that they are also differentiable. But it is also necessary to generalize the second fundamental form. Let it be: ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle N_{1},...,N_{k}}$

${\displaystyle l_{\sigma ,ij}=(X_{ij},N_{\sigma })}$

where so the equation applies first ${\displaystyle \sigma =1,...,k.}$

${\displaystyle X_{ij}=\Gamma _{ij}^{k}X_{k}+\sum _{\sigma =1}^{k}l_{\sigma ,ij}N_{\sigma }.}$

For the second part of the Gauss-Weingarten equations, the so-called torsion coefficients are required:

${\displaystyle T_{\sigma ,i}^{\vartheta }=(N_{\sigma ,i},N_{\vartheta })=-(N_{\sigma },N_{\vartheta ,i})=-T_{\vartheta ,i}^{\sigma }.}$

These quantities are comparable to the winding or torsion of curves. This gives for the second part of the Gauß-Weingarten equations:

${\displaystyle N_{\sigma ,i}=-l_{\sigma ,ij}g^{jk}X_{k}+\sum _{\vartheta =1}^{k}T_{\sigma ,i}^{\vartheta }N_{\vartheta }.}$

literature

• Wilhelm Blaschke : Lectures on differential geometry and geometric basics of Einstein's theory of relativity. Volume 1: Elementary Differential Geometry (= the basic teachings of the mathematical sciences in individual representations. With special consideration of the areas of application 1, ISSN  0072-7830 ). Springer, Berlin 1921, Paragraph 46, 48.
• Dirk J. Struik : Lectures on Classical Differential Geometry. 2nd Edition. Dover, New York NY 1961, pp. 106f.

Individual evidence

1. ^ Blaschke Lectures Differential Geometry , Volume 1, p. 78
2. ↑ Struik Lectures on Classical Differential Geometry , p. 108
3. ↑ Blaschke lectures on differential geometry , Volume 1, p. 75