Hyperfläche/Raum/Parametrisierung/Christoffelsymbole/Bezug zu riemannscher Metrik/Fakt/Beweis

Beweis

Es ist

${}{\begin{aligned}\partial _{k}g_{ij}&=\partial _{k}\left\langle \partial _{i}\varphi ,\partial _{j}\varphi \right\rangle \\&=\left\langle \partial _{k}\partial _{i}\varphi ,\partial _{j}\varphi \right\rangle +\left\langle \partial _{i}\varphi ,\partial _{k}\partial _{j}\varphi \right\rangle \\&=\left\langle \Gamma _{ki}^{1}\partial _{1}\varphi +\Gamma _{ki}^{2}\partial _{2}\varphi +h_{ki}N,\partial _{j}\varphi \right\rangle +\left\langle \partial _{i}\varphi ,\Gamma _{kj}^{1}\partial _{1}\varphi +\Gamma _{kj}^{2}\partial _{2}\varphi +h_{kj}N\right\rangle \\&=\left\langle \Gamma _{ki}^{1}\partial _{1}\varphi +\Gamma _{ki}^{2}\partial _{2}\varphi ,\partial _{j}\varphi \right\rangle +\left\langle \partial _{i}\varphi ,\Gamma _{kj}^{1}\partial _{1}\varphi +\Gamma _{kj}^{2}\partial _{2}\varphi \right\rangle \\&=\Gamma _{ki}^{1}g_{1j}+\Gamma _{ki}^{2}g_{2j}+\Gamma _{kj}^{1}g_{i1}+\Gamma _{kj}^{2}g_{i2}.\end{aligned}}$

Somit ist

${}{\begin{aligned}&\,\,\,\,\,\,\,\partial _{i}g_{jk}+\partial _{j}g_{ki}-\partial _{k}g_{ij}\\&=\Gamma _{ij}^{1}g_{1k}+\Gamma _{ij}^{2}g_{2k}+\Gamma _{ik}^{1}g_{j1}+\Gamma _{ik}^{2}g_{j2}+\Gamma _{jk}^{1}g_{1i}+\Gamma _{jk}^{2}g_{2i}+\Gamma _{ji}^{1}g_{k1}+\Gamma _{ji}^{2}g_{k2}-{\left(\Gamma _{ki}^{1}g_{1j}+\Gamma _{ki}^{2}g_{2j}+\Gamma _{kj}^{1}g_{i1}+\Gamma _{kj}^{2}g_{i2}\right)}\\&=2{\left(\Gamma _{ij}^{1}g_{1k}+\Gamma _{ij}^{2}g_{2k}\right)}.\,\end{aligned}}$

Es liegt also die Matrixbeziehung

${}{\begin{pmatrix}\partial _{i}g_{j1}+\partial _{j}g_{1i}-\partial _{1}g_{ij}\\\partial _{i}g_{j2}+\partial _{j}g_{2i}-\partial _{2}g_{ij}\end{pmatrix}}=2{\begin{pmatrix}g_{11}&g_{12}\\g_{12}&g_{22}\end{pmatrix}}{\begin{pmatrix}\Gamma _{ij}^{1}\\\Gamma _{ij}^{2}\end{pmatrix}}\,$

vor, woraus durch Multiplikation mit ${}{\frac {1}{2}}G^{-1}$ die Behauptung folgt.

Zur bewiesenen Aussage