3. Differential equations and curvatures
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 \( {\boldsymbol g}^{-1}. \) Setting \[ D\psi^{-1} =:\left( \begin{array}{cc} \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}1} & \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}1} \\[0.6ex] \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}2} & \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}2} \end{array} \right),\quad (D\psi^{-1})^T =:\left( \begin{array}{cc} \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}1} & \overline\Lambda{\ }_{\mspace{-0.4ex}1}^{\mspace{-0.4ex}2} \\[0.6ex] \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}1} & \overline\Lambda{\ }_{\mspace{-0.4ex}2}^{\mspace{-0.4ex}2} \end{array} \right), \] 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 \[ \overset{\boldsymbol{\!\sim}}\Gamma{\ }_{\mspace{-0.4ex}ij}^{\mspace{-0.4ex}k} \] 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} Third line: We compute \[ \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} \]