3. Differential equations
3.1.1 The inverse first fundamental form
Next, we want to proceed with introducing the coefficients \( g^{ij} \) of the inverse first fundamental form of a regular surface \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}), \) i.e. \[ \boldsymbol{g}^{-1} =\left( \begin{array}{cc} g_{11} & g_{12} \\ g_{21} & g_{22} \end{array} \right) =:\left( \begin{array}{cc} g^{11} & g^{12} \\ g^{21} & g^{22} \end{array} \right) \] with the characteristic property \[ \sum_{j=1}^2g_{ij}g^{jk}=\delta_i^k\,, \] where \( \delta_i^k \) is the known Kronecker symbol we introduced → here.
3.1.2 Transformation behaviour of the inverse first fundamental form I
From → this paragraph we recall the transformation formula \[ \overset{\boldsymbol{\,\sim}}{\boldsymbol g}=D\psi^T\circ{\boldsymbol g}\circ D\psi \] for the first fundamental form \( {\boldsymbol g}. \) From this rule we immediately get \[ \overset{\boldsymbol{\,\sim}}{\boldsymbol g}^{-1} =(D\psi)^{-1}\circ{\boldsymbol g}^{-1}\circ(D\psi^T)^{-1} =(D\psi)^{-1}\circ{\boldsymbol g}^{-1}\circ(D\psi^{-1})^T\,. \] This is the transformation formula for the inverse first fundamental form \( {\boldsymbol g}^{-1}. \) Let again \[ D\psi^{-1} =\left( \begin{array}{cc} \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}1} & \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}1} \\[0.6ex] \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}2} & \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}2} \end{array} \right),\quad (D\psi^{-1})^T =\left( \begin{array}{cc} \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}1} & \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}2} \\[0.6ex] \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}1} & \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}2} \end{array} \right), \] where the coefficients \( \overline\Lambda{\ }_{\mspace{-0.4ex}i}^{\mspace{-0.4ex}j} \) satisfy \[ \sum_{j=1}^2\Lambda_j^i\overline\Lambda{\ }_{\mspace{-0.4ex}k}^{\mspace{-0.4ex}j} =\sum_{j=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}j}^{\mspace{-0.4ex}i}\Lambda_k^j=\delta_k^i\,,\quad i,k=1,2, \] we can rewrite the above result into the following index form.
Proposition: It holds the transformation formula \[ \overset{\boldsymbol{\,\sim}}g{\ }^{\mspace{-0.3ex}mn} =\sum_{i,j=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}i}^{\mspace{-0.4ex}m}\overline\Lambda{\ }_{\mspace{-0.4ex}j}^{\mspace{-0.4ex}n}g^{ij}\,,\quad m,n=1,2, \] for the inverse first fundamental form.
3.1.3 Transformation behaviour of the inverse first fundamental form II
We want to present a tensor calculus derivation of the transformation formula of the inverse first fundamental. For this purpose, we recall the first identity from → this paragraph and start as follows \begin{align} \overset{\boldsymbol{\,\sim}}g{\ }^{\!mn} &= \sum_{i,j=1}^2\overset{\boldsymbol{\,\sim}}g_{ij}\overset{\boldsymbol{\,\sim}}g{\ }^{\!im}\overset{\boldsymbol{\,\sim}}g{\ }^{\!jn} \,=\,\sum_{i,j=1}^2\sum_{k,\ell=1}^2\Lambda_i^k\Lambda_j^\ell g_{k\ell}\overset{\boldsymbol{\,\sim}}g{\ }^{\!im}\overset{\boldsymbol{\,\sim}}g{\ }^{\!jn} \\[1ex] &= \sum_{i,j=1}^2\sum_{k,\ell=1}^2(\Lambda_i^k\overset{\boldsymbol{\,\sim}}g{\ }^{\!im})(\Lambda_j^\ell\overset{\boldsymbol{\,\sim}}g{\ }^{\!jn})g_{k\ell} \\[1ex] &= \sum_{i,j=1}^2\sum_{k,\ell=1}^2(\Lambda_i^k\overset{\boldsymbol{\,\sim}}g{\ }^{\!im})(\Lambda_j^\ell\overset{\boldsymbol{\,\sim}}g{\ }^{\!jn})\sum_{r,s=1}^2g_{kr}g_{\ell s}g^{rs} \\[1ex] &= \sum_{r,s=1}^2 \left(\,\sum_{i,k=1}^2\Lambda_i^k\overset{\boldsymbol{\,\sim}}g{\ }^{\!im}g_{kr}\right) \left(\,\sum_{j,\ell=1}^2\Lambda_j^\ell\overset{\boldsymbol{\,\sim}}g{\ }^{\!jn}g_{\ell s}\right)g^{rs}\,. \end{align} Now, we multiply \( \overset{\boldsymbol{\,\sim}}g_{mn} \) from → here by \( \overset{\boldsymbol{\!\!\!\sim}}{\Lambda_k^m} \) and sum over \( m \) to obtain \[ \sum_{m=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}k}^{\mspace{-0.4ex}m}\overset{\boldsymbol{\,\sim}}g_{mn} =\sum_{i,j=1}^2\sum_{m=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}k}^{\mspace{-0.4ex}m}\Lambda_m^i\Lambda_n^jg_{ij} =\sum_{i,j=1}^2\delta_k^i\Lambda_n^jg_{ij} =\sum_{j=1}^2\Lambda_n^jg_{kj}\,, \] and a second multiplication by \( \overline\Lambda{\ }_{\mspace{-0.4ex}\ell}^{\mspace{-0.4ex}n} \) and summation over \( n \) yields \[ \sum_{m,n=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}k}^{\mspace{-0.4ex}m}\overline\Lambda{\ }_{\mspace{-0.4ex}\ell}^{\mspace{-0.4ex}n}\overset{\boldsymbol{\,\sim}}g_{mn} =\sum_{j,n=1}^2\Lambda_n^j\overline\Lambda{\ }_{\mspace{-0.4ex}\ell}^{\mspace{-0.4ex}n} g_{kj} =\sum_{j=1}^2\delta_\ell^jg_{kj} =g_{k\ell}\,. \] We substitute this identity for \( g_{kr} \) and \( g_{\ell s} \) in the right hand side of the above result and arrive at \begin{align} \overset{\boldsymbol{\,\sim}}g{\ }^{\!mn} &= \sum_{r,s=1}^2 \left(\, \sum_{i,k=1}^2\sum_{a,b=1}^2 \Lambda_i^k\overset{\boldsymbol{\,\sim}}g{\ }^{\!im} \overline\Lambda{\ }_{\mspace{-0.4ex}k}^{\mspace{-0.4ex}a}\overline\Lambda{\ }_{\mspace{-0.4ex}r}^{\mspace{-0.4ex}b}\overset{\boldsymbol{\,\sim}}g_{ab} \right) \left(\, \sum_{j,\ell=1}^2\sum_{a,b=1}^2 \Lambda_j^\ell\overset{\boldsymbol{\,\sim}}g{\ }^{\!jn} \overline\Lambda{\ }_{\mspace{-0.4ex}\ell}^{\mspace{-0.4ex}c}\overline\Lambda{\ }_{\mspace{-0.4ex}s}^{\mspace{-0.4ex}d}\overset{\boldsymbol{\,\sim}}g_{cd} \right) g^{rs} \\[1.6ex] &= \sum_{r,s=1}^2 \left(\, \sum_{i=1}^2\sum_{a,b=1}^2 \delta_i^a\overset{\boldsymbol{\,\sim}}g{\ }^{\!im} \overline\Lambda{\ }_{\mspace{-0.4ex}r}^{\mspace{-0.4ex}b}\overset{\boldsymbol{\,\sim}}g_{ab} \right) \left(\, \sum_{j=1}^2\sum_{c,d=1}^2 \delta_j^c\overset{\boldsymbol{\,\sim}}g{\ }^{\!jn} \overline\Lambda{\ }_{\mspace{-0.4ex}s}^{\mspace{-0.4ex}d}\overset{\boldsymbol{\,\sim}}g_{cd} \right) g^{rs} \\[1.6ex] &= \sum_{r,s=1}^2 \left(\, \sum_{i=1}^2\sum_{b=1}^2 \overset{\boldsymbol{\,\sim}}g{\ }^{\!im} \overline\Lambda{\ }_{\mspace{-0.4ex}r}^{\mspace{-0.4ex}b}\overset{\boldsymbol{\,\sim}}g_{ib} \right) \left(\, \sum_{j=1}^2\sum_{d=1}^2 \overset{\boldsymbol{\,\sim}}g{\ }^{\!jn} \overline\Lambda{\ }_{\mspace{-0.4ex}s}^{\mspace{-0.4ex}d}\overset{\boldsymbol{\,\sim}}g_{jd} \right) g^{rs} \\[1.6ex] &= \sum_{r,s=1}^2 \left(\, \sum_{b=1}^2\delta_b^m\overline\Lambda{\ }_{\mspace{-0.4ex}r}^{\mspace{-0.4ex}b} \right) \left(\, \sum_{d=1}^2\delta_d^n\overline\Lambda{\ }_{\mspace{-0.4ex}s}^{\mspace{-0.4ex}d} \right) g^{rs} \\[1.6ex] &= \sum_{r,s=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}r}^{\mspace{-0.4ex}m}\overline\Lambda{\ }_{\mspace{-0.4ex}s}^{\mspace{-0.4ex}n}g^{rs}\,. \end{align} This is our second proof of the transformation formula for \( \overset{\boldsymbol{\,\sim}}g{\ }^{\!mn} \) from the preceding paragraph.\( \qquad\Box \)
Definition: The Christoffel symbols of a regular surface \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) are defined as \[ \Gamma_{ij}^k:=\frac{1}{2}\,\sum_{\ell=1}^2g^{k\ell}(g_{j\ell,u^i}+g_{\ell i,u^j}-g_{ij,u^\ell}) \] for \( i,j,k=1,2, \) where the lower index \( u^j \) denotes the partial derivative w.r.t. \( u^j. \)
Note the symmetry of the Christoffel symbols \[ \Gamma_{ij}^k=\Gamma_{ji}^k\,,\quad i,j,k=1,2. \] Finally, we want to write down them using conformal parameters \[ g_{11}=g_{22}=W,\quad g_{12}=0 \quad\mbox{in}\ B \] with the area element \( W, \) namely \begin{align} & \Gamma_{11}^1=\frac{W_u}{2W}\,,\quad \Gamma_{12}^1=\Gamma_{21}^1=\frac{W_v}{2W}\,,\quad \Gamma_{22}^1=-\,\frac{W_u}{2W}\,, \\[1ex] & \Gamma_{11}^2=-\,\frac{W_v}{2W}\,,\quad \Gamma_{12}^2=\Gamma_{21}^2=\frac{W_u}{2W}\,,\quad \Gamma_{22}^2=\frac{W_v}{2W}\,. \end{align}
3.1.5 Transformation behaviour of the Christoffel symbols
Proposition: It holds the transformation formula \[ \overset{\boldsymbol{\!\sim}}\Gamma{\ }_{\mspace{-0.4ex}ij}^{\mspace{-0.4ex}k} =\sum_{a,b,c=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}c}^{\mspace{-0.4ex}k}\Lambda_i^a\Lambda_j^b\Gamma_{ab}^c +\sum_{m=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\Lambda_{j,\overset{\boldsymbol{\sim}}{u}^i}^m\,, \quad i,j,k=1,2, \] for the Christoffel symbols.
Proof: Let the symbol \( \partial_{u^i} \) denote the partial derivative w.r.t. \( u^i. \) We calculate \begin{align} \overset{\boldsymbol{\!\sim}}\Gamma{\ }_{\mspace{-0.4ex}ij}^{\mspace{-0.4ex}k} &= \frac{1}{2}\,\sum_{\ell=1}^2 \overset{\boldsymbol{\,\sim}}g{\ }^{\mspace{-0.3ex}k\ell} \big( \overset{\boldsymbol{\,\sim}}g_{j\ell,\overset{\boldsymbol{\sim}}u{\ }^{\mspace{-0.3ex}i}} +\overset{\boldsymbol{\,\sim}}g_{\ell i,\overset{\boldsymbol{\sim}}u{\ }^{\mspace{-0.3ex}j}} -\overset{\boldsymbol{\,\sim}}g_{ij,\overset{\boldsymbol{\sim}}u{\ }^{\mspace{-0.3ex}\ell}} \big) \\[1ex] &= \frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\partial_{u^c}\Lambda_j^a\Lambda_\ell^bg_{ab}+\Lambda_j^c\partial_{u^c}\Lambda_\ell^a\Lambda_i^bg_{ab}-\Lambda_\ell^c\partial_{u^c}\Lambda_i^a\Lambda_j^bg_{ab} \big) \\[1ex] &= \frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_{j,u^c}^a\Lambda_\ell^bg_{ab}+\Lambda_j^c\Lambda_{\ell,u^c}^a\Lambda_i^bg_{ab}-\Lambda_\ell^c\Lambda_{i,u^c}^a\Lambda_j^bg_{ab} \big) \\[1ex] & \phantom{=\,\,} +\frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_j^a\Lambda_{\ell,u^c}^bg_{ab}+\Lambda_j^c\Lambda_\ell^a\Lambda_{i,u^c}^bg_{ab}-\Lambda_\ell^c\Lambda_i^a\Lambda_{j,u^c}^bg_{ab} \big) \\[1ex] & \phantom{=\,\,} +\frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_j^a\Lambda_\ell^bg_{ab,u^c}+\Lambda_j^c\Lambda_\ell^a\Lambda_i^bg_{ab,u^c}-\Lambda_\ell^c\Lambda_i^a\Lambda_j^bg_{ab,u^c} \big) \end{align} We begin with the third line \[ \begin{array}{l} \displaystyle \frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_j^a\Lambda_\ell^bg_{ab,u^c}+\Lambda_j^c\Lambda_\ell^a\Lambda_i^bg_{ab,u^c}-\Lambda_\ell^c\Lambda_i^a\Lambda_j^bg_{ab,u^c} \big) \\[1ex] \quad\displaystyle =\,\frac{1}{2}\,\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{mn}\Lambda_i^c\Lambda_j^a\delta_n^bg_{ab,u^c} +\frac{1}{2}\,\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{mn}\Lambda_j^c\Lambda_i^b\delta_n^ag_{ab,u^c} \\[1ex] \qquad\displaystyle +\frac{1}{2}\,\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{mn}\Lambda_i^a\Lambda_j^b\delta_n^bg_{ab,u^c} \\[1ex] \quad\displaystyle =\,\frac{1}{2}\,\sum_{m=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{mb}\Lambda_i^c\Lambda_j^ag_{ab,u^c} +\frac{1}{2}\,\sum_{m=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{ma}\Lambda_j^c\Lambda_i^bg_{ab,u^c} \\[1ex] \qquad\displaystyle +\frac{1}{2}\,\sum_{m=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{mc}\Lambda_i^a\Lambda_j^bg_{ab,u^c} \\[1ex] \quad\displaystyle =\,\frac{1}{2}\,\sum_{m=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k} (g^{mb}\Lambda_i^c\Lambda_j^ag_{ab,u^c}+g^{ma}\Lambda_j^c\Lambda_i^bg_{ab,u^c}-g^{mc}\Lambda_i^a\Lambda_j^bg_{ab,u^c}) \\[1ex] \quad\displaystyle =\,\frac{1}{2}\,\sum_{m=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k} (g^{mc}\Lambda_i^a\Lambda_j^bg_{bc,u^a}+g^{mc}\Lambda_i^a\Lambda_j^bg_{ca,u^b}-g^{mc}\Lambda_i^a\Lambda_j^bg_{ab,u^c}) \\[1ex] \quad\displaystyle =\,\frac{1}{2}\,\sum_{m=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k} \Lambda_i^a\Lambda_j^bg^{cm}(g_{bc,u^a}+g_{ca,u^b}-g_{ab,u^c}) \\[1ex] \quad\displaystyle =\,\frac{1}{2}\,\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}c}^{\mspace{-0.4ex}k}\Lambda_i^a\Lambda_j^b \sum_{m=1}^2g^{cm}(g_{bm,u^a}+g_{ma,u^b}-g_{ab,u^m}) \\[1ex] \quad\displaystyle =\,\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}c}^{\mspace{-0.4ex}k}\Lambda_i^a\Lambda_j^b \Gamma_{ab}^c\,. \end{array} \] Next, we interchange \( a \) and \( b \) in the third line, and, furthermore, we can interchange partial derivatives due to our regularity assumption \( \psi\in C^{4+\alpha}(\overset{\boldsymbol{\,\sim}}{B},B), \) i.e. it holds \[ \sum_{c=1}^2\Lambda_i^c\Lambda_{j,u^c}^a=\sum_{c=1}^2\Lambda_j^c\Lambda_{i,u^c}^a\quad\text{etc.}\,, \] such that we arrive at \[ \begin{array}{l} \displaystyle \frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_{j,u^c}^a\Lambda_\ell^bg_{ab}+\Lambda_j^c\Lambda_{\ell,u^c}^a\Lambda_i^bg_{ab}-\Lambda_\ell^c\Lambda_{i,u^c}^a\Lambda_j^bg_{ab} \big) \\[1ex] \qquad\displaystyle +\frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_j^a\Lambda_{\ell,u^c}^bg_{ab}+\Lambda_j^c\Lambda_\ell^a\Lambda_{i,u^c}^bg_{ab}-\Lambda_\ell^c\Lambda_i^a\Lambda_{j,u^c}^bg_{ab} \big) \\[3ex] \quad\displaystyle =\frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_{j,u^c}^a\Lambda_\ell^bg_{ab}+\Lambda_j^c\Lambda_{\ell,u^c}^a\Lambda_i^bg_{ab}-\Lambda_\ell^c\Lambda_{i,u^c}^a\Lambda_j^bg_{ab} \big) \\[3ex] \qquad\displaystyle +\frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_j^b\Lambda_{\ell,u^c}^ag_{ab}+\Lambda_j^c\Lambda_\ell^b\Lambda_{i,u^a}^bg_{ab}-\Lambda_\ell^c\Lambda_i^b\Lambda_{j,u^c}^ag_{ab} \big) \\[3ex] \quad\displaystyle =\frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_i^c\Lambda_{j,u^c}^a\Lambda_\ell^bg_{ab}+\Lambda_j^c\Lambda_{\ell,u^c}^a\Lambda_i^bg_{ab}-\Lambda_\ell^c\Lambda_{i,u^c}^a\Lambda_j^bg_{ab} \big) \\[3ex] \qquad\displaystyle +\frac{1}{2}\,\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \big( \Lambda_\ell^c\Lambda_j^b\Lambda_{i,u^c}^ag_{ab}+\Lambda_i^c\Lambda_\ell^b\Lambda_{j,u^a}^bg_{ab}-\Lambda_j^c\Lambda_i^b\Lambda_{\ell,u^c}^ag_{ab} \big) \\[3ex] \quad\displaystyle =\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \Lambda_i^c\Lambda_\ell^b\Lambda_{j,u^c}^ag_{ab}\,. \end{array} \] We continue \[ \begin{array}{l} \displaystyle \sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}\ell}g^{mn} \Lambda_i^c\Lambda_\ell^b\Lambda_{j,u^c}g_{ab} \\[3ex] \quad\displaystyle =\sum_{\ell=1}^2\sum_{m,n=1}^2\sum_{a,b,c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{mn} \Lambda_i^c\Lambda_{j,u^c}\delta_n^bg_{ab} =\sum_{m,n=1}^2\sum_{a,c=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}g^{mn}\Lambda_i^c\Lambda_{j,u^c}^ag_{an} \\[3ex] \quad\displaystyle =\sum_{m,n=1}^2\sum_{a,c=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\Lambda_i^c\Lambda_{j,u^c}\delta_a^m =\sum_{m=1}^2\sum_{a,c=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\Lambda_i^c\Lambda_{j,u^c}^a\delta_a^m \\[3ex] \quad\displaystyle =\sum_{m=1}^2\sum_{c=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\Lambda_i^c\Lambda_{j,u^c}^m =\sum_{m=1}^2\sum_{c=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\Lambda_{j,\overset{\boldsymbol{\sim}}{u}^i}^m \end{array} \] considering \[ \sum_{m=1}^2\sum_{c=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k}\Lambda_i^c\Lambda_{j,u^c}^m =\sum_{m=1}^2\sum_{c=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k} \frac{\partial u^c}{\partial\overset{\boldsymbol{\sim}}{u}^i}\frac{\partial}{\partial u^c}\frac{\partial u^m}{\partial\overset{\boldsymbol{\sim}}{u}^j} =\sum_{m=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}k} \partial_{\overset{\boldsymbol{\sim}}{u}^i}\Lambda_j^m \] after interchanging partial derivatives. This proves the proposition.\( \qquad\Box \)
The Gauss equations contain the expansion of the second derivatives of a surface parametrization \( X \) in terms of its tangential parts and its normal parts. The tangential vectors \( X_{u^i} \) and the unit normal vectors \( N_\sigma \) of an ONF of \( X \) are considered as basis vectors of the embedding space \( \mathbb R^{n+2}. \)
Proposition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization and \( {\mathfrak N}=(N_1,\ldots,N_n) \) be an ONF of \( X. \) Then there hold \[ X_{u^iu^j}=\sum_{k=1}^2\Gamma_{ij}^kX_{u^i}+\sum_{\sigma=1}^n L_{ij,\sigma}N_\sigma\quad\text{in}\ B \] for \( i,j=1,2 \) with the coefficients \[ L_{\sigma,ij}:=L_{N_\sigma,ij}\,,\quad i,j=1,2,\ \sigma=1,\ldots,n, \] of the → second fundamental form of \( X \) w.r.t. \( N_\sigma. \)
Proof: We consider the ansatz \[ X_{u^iu^j}=\sum_{k=1}^2 a_{ij}^kX_{u^k}+\sum_{\sigma=1}^nb_{\sigma,ij}N_\sigma \] with unknown functions \( a_{ij}^k \) and \( b_{\sigma,ij}. \) Multiplication by \( N_\omega\in{\mathfrak N} \) yields \[ L_{\omega,ij} =-\langle X_{u^i},N_{\omega,u^j}\rangle =\langle X_{u^iu^j},N_\omega\rangle =\sum_{\sigma=1}^nb_{\sigma,ij}\langle N_\sigma,N_\omega\rangle =b_{\omega,ij}\,. \] To determine the \( a_{ij}^k \) we multiply by \( X_{u^\ell} \) and arrive at \[ \langle X_{u^iu^j},X_{u^\ell}\rangle =\sum_{k=1}^2a_{ij}^k\langle X_{u^k},X_{u^\ell}\rangle =\sum_{k=1}^2a_{ij}^kg_{k\ell} =:a_{i\ell j}\,. \] Note that \( a_{i\ell j}=a_{j\ell i} \) due to \( X_{u^iu^j}=X_{u^ju^i}. \) We calculate \[ a_{i\ell j} =\langle X_{u^i},X_{u^\ell}\rangle_{u^j}-\langle X_{u^i},X_{u^\ell u^j}\rangle =g_{i\ell,u^j}-a_{\ell ij} \] and, therefore, \[ g_{i\ell,u^j}=a_{i\ell j}+a_{\ell ij}\,. \] It follows \[ g_{j\ell,u^i}+g_{\ell i,u^j}-g_{ij,u^\ell} =a_{j\ell i}+a_{\ell ji}+a_{\ell ij}+a_{i\ell j}-a_{ij\ell}-a_{ji\ell} =2a_{i\ell j} \] such that we conclude with the identity for \( a_{i\ell j} \) from above \[ a_{i\ell j} =\sum_{k=1}^2a_{ij}^kg_{k\ell} =\frac{1}{2}\,(g_{j\ell,u^i}+g_{\ell i,u^j}-g_{ij,u^\ell}). \] Rearranging gives \[ a_{ij}^m =\frac{1}{2}\,\sum_{\ell=1}^2g^{m\ell}(g_{j\ell,u^i}+g_{\ell i,u^j}-g_{ij,u^\ell})=\Gamma_{ij}^m\,. \] with the Christoffel symbols \( \Gamma_{ij}^m. \) This proves the statement.\( \qquad\Box \)
This proof follows → W. Blaschke, K. Leichtweiß 1973.
3.2.1 The torsion coefficients
Definition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization and \( {\mathfrak N}=(N_1,\ldots,N_n) \) be an ONF of \( X. \) The torsion coefficients of \( {\mathfrak N} \) are defined as \[ T_{\sigma,i}^\vartheta:=\langle N_{\sigma,u^i},N_\vartheta\rangle \] for \( i=1,2 \) and \( \sigma,\vartheta=1,\ldots,n. \)
Remark: Note that the torsion coefficients \( T_{\sigma,i}^\vartheta \) are skew symmetric w.r.t. interchanging \( \sigma \) and \( \vartheta, \) since differentiating \( 0=\langle N_\sigma,N_\vartheta\rangle \) w.r.t. \( u^i \) yields \[ 0=\langle N_{\sigma,u^i},N_\vartheta\rangle+\langle N_\sigma,N_{\vartheta,u^i}\rangle=T_{\sigma,i}^\vartheta+T_{\vartheta,i}^\sigma \] and, therefore, \[ T_{\sigma,i}^\vartheta=-T_{\vartheta,i}^\sigma\quad\mbox{for all}\ i=1,2,\ \sigma,\vartheta=1,\ldots,n. \] The torsion vanishes identically in case of one codimension \( n=1. \)
Many of our definitions and identities can be found in well-known textbooks on differential geometry, for example → B.-Y. Chen 1973, → H. Brauner 1981, or → M.P. do Carmo 1992. The terminology torsion essentially goes back to → H. Weyl (translated by me):
From a normal vector \( {\mathbf n} \) in \( P \) there arises a vector \( {\mathbf n}'+d{\mathbf t} \) (\( {\mathbf n}' \) normal, \( d{\mathbf t} \) tangential). The infinitesimal linear mapping \( {\mathbf n}\mapsto{\mathbf n}' \) from \( {\mathfrak N}_P \) to \( {\mathfrak N}_{P'} \) is the torsion.
3.2.2 Transformation behaviour of the second fundamental form
Now we consider regular parameter transformations which leave the unit normal vectors of a given ONF unchanged, i.e. \[ \overset{\boldsymbol{\,\sim}}{N}_\sigma(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) =N_\sigma(u(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}),v(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})),\quad\sigma=1,\ldots,n. \]
Proposition: It holds the transformation formula \[ \overset{\boldsymbol{\!\sim}}L_{\sigma,mn}=\sum_{i,j=1}^2\Lambda_m^i\Lambda_n^jL_{\sigma,ij}\,,\quad m,n=1,2,\ \sigma=1,\ldots,n. \]
Proof: We start with \[ \overset{\boldsymbol{\,\sim}}{X}_{\overset{\boldsymbol{\sim}}{u}^m}=\sum_{i=1}^2\Lambda_m^iX_{u^i} \] and, therefore, \[ \overset{\boldsymbol{\,\sim}}{X}_{\overset{\boldsymbol{\sim}}{u}^m\overset{\boldsymbol{\sim}}{u}^n} =\sum_{i=1}^2\sum_{r=1}^2\left\{\Lambda_{m,u^r}^i\Lambda_n^r X_{u^i}+\Lambda_m^i\Lambda_n^rX_{u^iu^r}\right\}. \] A multiplication by \( \overset{\boldsymbol{\,\sim}}{N}_\sigma \) yields \begin{align} \langle \overset{\boldsymbol{\,\sim}}{X}_{\overset{\boldsymbol{\sim}}{u}^m\overset{\boldsymbol{\sim}}{u}^n},\overset{\boldsymbol{\,\sim}}{N}_\sigma\rangle &= \sum_{i=1}^2\sum_{r=1}^2 \left\{ \Lambda_{m,u^r}^i\Lambda_n^r\langle X_{u^i},\overset{\boldsymbol{\,\sim}}{N}_\sigma\rangle +\Lambda_m^i\Lambda_n^r\langle X_{u^iu^r},\overset{\boldsymbol{\,\sim}}{N}_\sigma\rangle \right\} \\[1ex] &= \sum_{i=1}^2\sum_{r=1}^2 \bigg\{ \Lambda_{m,u^r}^i\Lambda_n^r\langle X_{u^i},N_\sigma\rangle +\Lambda_m^i\Lambda_n^r\langle X_{u^iu^r},N_\sigma\rangle \bigg\} \\[1ex] &= \sum_{i=1}^2\sum_{r=1}^2\Lambda_m^i\Lambda_n^r\langle X_{u^iu^r},N_\sigma\rangle =\sum_{i=1}^2\sum_{r=1}^2\Lambda_m^i\Lambda_n^rL_{\sigma,ir}\,. \end{align} This proves the proposition.\( \qquad\Box \)
3.2.3 Transformation behaviour of the torsion coefficients
We consider again regular parameter transformations which leave the unit normal vectors of a given ONF unchanged, i.e. \[ \overset{\boldsymbol{\,\sim}}{N}_\sigma(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) =N_\sigma(u(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}),v(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})),\quad\sigma=1,\ldots,n. \]
Proposition: It holds the transformation formula \[ \overset{\boldsymbol{\!\sim}}T{\ }_{\mspace{-0.4ex}\sigma,m}^{\mspace{-0.4ex}\vartheta} =\sum_{i=1}^2\Lambda_m^iT_{\sigma,i}^\vartheta\,,\quad m=1,2,\ \sigma,\vartheta=1,\ldots,n. \]
Proof: We compute \[ \overset{\boldsymbol{\!\sim}}T{\ }_{\mspace{-0.4ex}\sigma,m}^{\mspace{-0.4ex}\vartheta} =\langle\overset{\boldsymbol{\,\sim}}{N}_{\sigma,\overset{\boldsymbol{\sim}}{u}^m},\overset{\boldsymbol{\,\sim}}{N}_{\vartheta,}\rangle =\langle\Lambda_m^iN_{\sigma,u^i},N_\vartheta\rangle =\Lambda_m^iT_{\sigma,i}^\vartheta\,. \] This proves the proposition.\( \qquad\Box \)
3.2.4 The Weingarten equations
The Weingarten equations contain the expansion of the first derivatives of an unit normal vector of a surface parametrization \( X \) in terms of its tangential parts and its normal parts. The tangential vectors \( X_{u^i} \) and the unit normal vectors \( N_\sigma \) of an ONF of \( X \) are considered as basis vectors of the embedding space \( \mathbb R^{n+2}. \)
Proposition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization and \( {\mathfrak N}=(N_1,\ldots,N_n) \) be an ONF of \( X. \) Then there hold \[ N_{\sigma,u^i}=-\sum_{k=1}^2L_{\sigma,ij}g^{jk}X_{u^k}+\sum_{\vartheta=1}^nT_{\sigma,i}^\vartheta\,,\quad\text{in}\ B \] for \( i=1,2 \) and \( \sigma=1,\ldots,n. \)
Proof: With unknown functions \( a_{\sigma,i}^k \) and \( b_{\sigma,i}^\vartheta \) we evaluate the ansatz \[ N_{\sigma,u^i}=\sum_{k=1}^2a_{\sigma,i}^kX_{u^k}+\sum_{\vartheta=1}^nb_{\sigma,i}^\vartheta N_\vartheta\,,\quad i=1,2,\ \sigma=1,\ldots,n. \] Multiplication by \( X_{u^\ell} \) brings \[ -L_{\sigma,i\ell} =\langle N_{\sigma,u^i},X_{u^\ell}\rangle =\sum_{k=1}^2a_{\sigma,i}^k\langle X_{u^k},X_{u^\ell}\rangle =\sum_{k=1}^2a_{\sigma,i}^kg_{k\ell}\,, \] and rearranging yields \[ a_{\sigma,i}^m =\sum_{k=1}^2a_{\sigma,i}^k\delta_k^m =\sum_{k,\ell=1}^2a_{\sigma,i}^kg_{k\ell}g^{\ell m} =-\sum_{\ell=1}^2L_{\sigma,i\ell}g^{\ell m}\,. \] Next, a multiplication of our ansatz by \( N_\omega \) gives \[ T_{\sigma,i}^\omega =\langle N_{\sigma,u^i},N_\omega\rangle =\sum_{\vartheta=1}^nb_{\sigma,i}^\vartheta\langle N_\vartheta,N_\omega\rangle =\sum_{\vartheta=1}^nb_{\sigma,i}^\vartheta\delta_{\vartheta\omega} =b_{\sigma,i}^\omega\,. \] Inserting our results for \( a_{\sigma,i}^m \) und \( b_{\sigma,i}^\omega \) into the ansatz proves the proposition.\( \qquad\Box \)
This proof follows → W. Blaschke, K. Leichtweiß 1973.