The fundamental equations of surface theory describe how a surface bends and stretches in three-dimensional space. The second fundamental form of a surface is represented again, which relate the second partial derivatives of the position vector \(\vec{r}\) with respect to the surface parameters \(u\) and \(v\) to the coefficients of curvature and the normal vector \(\vec{N}\). The equation for the second partial derivative with respect to \(u\) is: \( \vec{r}_{11} = L\vec{N} + l\vec{r}_1 + \lambda \vec{r}_2 \). Here, \(L\) represents the coefficient of the second fundamental form in the \(u\) direction, indicating the normal curvature in the \(u\)-direction. The terms \(l\vec{r}_1\) and \(\lambda \vec{r}_2\) account for the tangential curvature components, where \(l\) and \(\lambda\) are coefficients reflecting how the surface stretches or compresses in this direction. Similarly, other two equations. These equations together provide a detailed description of the surface's extrinsic curvature and local behavior, integrating how the surface bends and stretches in three-dimensional space, given as below.
Let\( S:\vec{r}=\vec{r}( u,v )\) be a surface with fundamental coefficients
\( E=\vec{r}_1^2,F=\vec{r}_1.\vec{r}_2,G=\vec{r}_2^2\)
\( L=\vec{r}_{11}.\vec{N},M=\vec{r}_{12}.\vec{N},N=\vec{r}_{22}.\vec{N}\)
Now, three equations relating fundamental coefficients and their derivatives, as given below, are called fundamental equation of surface theory. It is given by
\( \vec{r}_{11}=L\vec{N}+l\vec{r}_1+\lambda \vec{r}_2 \)
\( \vec{r}_{12}=M\vec{N}+m\vec{r}_1+\mu \vec{r}_2 \)
\( \vec{r}_{22}=N\vec{N}+n\vec{r}_1+\nu \vec{r}_2 \)
In the matrix form, it is
\( \left[ \begin{matrix} \vec{r}_{11} \\ \vec{r}_{12} \\ \vec{r}_{22} \end{matrix} \right] = \left[ \begin{matrix} L & l & \lambda \\ M & m & \mu \\ N & n & \nu \end{matrix} \right] \left[ \begin{matrix} \vec{N} \\ \vec{r}_1 \\ \vec{r}_2 \end{matrix} \right] \)
Here,\( l,m,n,\lambda ,\mu ,\nu \) are called christoffel coefficients.
Relation of Fundamental Coefficients and their derivatives
\(\vec{r}_{11}\cdot \vec{r}_1=\frac{1}{2} E_1 \)\(\vec{r}_{22}\cdot \vec{r}_2=\frac{1}{2} G_2 \)
\(\vec{r}_{12}\cdot \vec{r}_1=\frac{1}{2}E_2 \)
\(\vec{r}_{12}\cdot \vec{r}_2=\frac{1}{2} G_1 \)
\(\vec{r}_{11}\cdot \vec{r}_2=F_1-\frac{1}{2}E_2 \)
\(\vec{r}_{22}\cdot \vec{r}_1=F_2-\frac{1}{2} G_1 \)
Christoffel Cofficient
Let \( S:\vec{r}=\vec{r}( u,v )\) be a surface with fundamental equation\(\vec{r}_{11}=L\vec{N}+l\vec{r}_1+\lambda \vec{r}_2 \) (i)
\( \vec{r}_{12}=M\vec{N}+m\vec{r}_1+\mu \vec{r}_2 \) (ii)
\(\vec{r}_{22}=N\vec{N}+n\vec{r}_1+\nu \vec{r}_2 \) (iii)
Here, \(l,m,n,\lambda ,\mu ,\nu \) are christoffel coefficients.
To find the Christoffel coefficients \( l \) and \( \lambda \), we take dot product on both side of (i) by \( \vec{r}_1\), we get
\( \vec{r}_{11}=L\vec{N}+l\vec{r}_1+\lambda \vec{r}_2 \)
or\( \vec{r}_{11}. \vec{r}_1=L\vec{N}.\vec{r}_1+l\vec{r}_1.\vec{r}_1+\lambda \vec{r}_2.\vec{r}_1 \)
or\( \vec{r}_{11}. \vec{r}_1=El+F\lambda \)
or\( \frac{1}{2} E_1=El+F\lambda \)
or\( El+F\lambda-\frac{1}{2} E_1=0 \) (A)
Again, we take dot product on both side of (i) by \( \vec{r}_2\), we get
\( \vec{r}_{11}=L\vec{N}+l\vec{r}_1+\lambda \vec{r}_2 \)
or\( \vec{r}_{11}. \vec{r}_2=L\vec{N}.\vec{r}_2+l\vec{r}_1.\vec{r}_2+\lambda \vec{r}_2.\vec{r}_2 \)
or\( \vec{r}_{11}. \vec{r}_2=Fl+G\lambda \)
or\( F_1-\frac{1}{2}E_2=Fl+G\lambda \)
or\( Fl+G\lambda -F_1+\frac{1}{2}E_2=0 \) (B)
Solving (A) and (B), for \( l\) and \( \lambda \), we get
\( l=\frac{GE_1-2FF_1+FE_2}{2H^2} \)
\( \lambda =\frac{2EF_1-EE_2-FE_1}{2H^2}\)
Similarly, we get
\( m=\frac{GE_2-FG_1}{2H^2} \)
\( \mu =\frac{EG_1-FE_2}{2H^2} \)
\( n=\frac{2GF_2-FG_2-GG_1}{2H^2} \)
\( \nu =\frac{FG_1+EG_2-2FF_2}{2H^2} \)
Gauss Characteristic Equation
In a surface, prove that\( \frac{1}{2}( 2F_{12}-E_{22}-G_{11})=( LN-M^2 )-( m^2-ln )E+( lv-2m\mu +\lambda n )F-( \mu ^2-\lambda v )G\)
We know that
\( E_{22}=2\vec{r}_{12}^2+2\vec{r}_2 \cdot \vec{r}_{122}\)
\(G_{11}=2\vec{r}_{12}^2+2\vec{r}_2 \cdot \vec{r}_{112}\)
\(F_{12}=\vec{r}_{11} \cdot \vec{r}_{22}+\vec{r}_{12} \cdot \vec{r}_{22}+\vec{r}_{12}^2+\vec{r}_1\cdot \vec{r}_{122}\)
Therefore, we have
\(2F_{12}-E_{22}-G_{11}=2( \vec{r}_{11} \cdot \vec{r}_{22}-\vec{r}_{12}^2 )\)
or\(\vec{r}_{11} \cdot \vec{r}_{22}-\vec{r}_{12}^2=\frac{1}{2}( 2F_{12}-E_{22}-G_{11} )\) (A)
Here, Gauss’s Formulas, we get
\(\vec{r}_{11}\cdot \vec{r}_{22}=( L \vec{N}+l\vec{r}_1+\lambda \vec{r}_2 )\cdot ( N \vec{N}+n\vec{r}_1+\nu \vec{r}_2)\)
\(\vec{r}_{11}\cdot \vec{r}_{22}=LN+lnE+( lv+\lambda n )F+\lambda vG\) (B)
Again, using Gauss’s Formulas, we get
\(\vec{r}_{12}^2=( M\vec{N}+m\vec{r}_1+\mu \vec{r}_2 )^2\)
\(\vec{r}_{12}^2=M^2+m^2 E+2m \mu F+ \mu ^2G\) (C)
Substituting (B) and (C) in (A), we get
\(\vec{r}_{11} \cdot \vec{r}_{22}-\vec{r}_{12}^2=\frac{1}{2}( 2F_{12}-E_{22}-G_{11} )\)
\(\frac{1}{2}( 2F_{12}-E_{22}-G_{11} )=\)
\((LN+lnE+( lv+\lambda n )F+\lambda vG)-( M^2+m^2E+2m\mu F+\mu ^2G )\)
\(\frac{1}{2}( 2F_{12}-E_{22}-G_{11} )=\)
\(( LN-M^2 )-( m^2-\ln )E+( lv-2m\mu +\lambda n )F-( \mu ^2-\lambda v )G\)
This completes the proof.
Mainardi-Codazzi Equation
The Mainardi-Codazzi equations express the relationships between the coefficients of the second fundamental form. The terms involved reflect how changes in the surface curvature interact with the surface’s intrinsic geometry.The equations are as below.\(L_2-M_1=mL-( l-\mu )M-\lambda N\)
\(M_2-N_1=nL-( m-\nu )M-\mu N\)
Proof
We consider the quantity
\( \frac{\partial }{\partial v}( \vec{r}_{11} )=\frac{\partial }{\partial u}( \vec{r}_{12} )\) (i)
Substituting the Gauss’s Formulas, we get
\(\frac{\partial }{\partial v}( L\vec{N}+l\vec{r}_1+\lambda \vec{r}_2 )=\frac{\partial }{\partial u}( M \vec{N}+m\vec{r}_1+\mu \vec{r}_2 )\)
Differentiating, we get
\(L_2 \vec{N}+L\vec{N}_2+l_2\vec{r}_1+l\vec{r}_{12}+\lambda _2 \vec{r}_2+\lambda \vec{r}_{22}=\)
\(M_1\vec{N}+M\vec{N}_1+m_1\vec{r}_1+m\vec{r}_{11}+\mu _1 \vec{r}_2+\mu \vec{r}_{12}\)
Operating dot product on both sides by \(\vec{N}\), we get
\(L_2+lM+\lambda N=M_1+mL+\mu M\)
or\(L_2-M_1=mL-( l-\mu )M-\lambda N\) (A)
For the next relation, we consider the quantity
\(\frac{\partial }{\partial v}( \vec{r}_{12} )=\frac{\partial }{\partial u}( \vec{r}_{22} )\) (ii)
Now substituting the Gauss’s Formulas, we get
\(\frac{\partial }{\partial v}( M\vec{N} +m\vec{r}_1+\mu \vec{r}_2)=\frac{\partial }{\partial u}( N\vec{N} +n\vec{r}_1+\nu \vec{r}_2 )\)
or\(M_2 \vec{N}+M \vec{N}_2+m_2\vec{r}_1+m\vec{r}_{12}+\mu _2\vec{r}_2+\mu \vec{r}_{22}=\)
\( N_1 \vec{N}+N \vec{N}_1+n_1\vec{r}_1+n\vec{r}_{11}+\nu _1\vec{r}_2+\nu \vec{r}_{12}\)
Operating dot product on both sides by \(\vec{N}\), we get
\(M_2+mM+\mu N=N_1+nL+\nu M\)
or\(M_2-N_1=nL-( m-\nu )M-\mu N\) (B)
From (A) and (B), the theorem completes.
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