Hyperfläche/Raum/Parametrisierung/Christoffelsymbole/Bezug zu riemannscher Metrik/Matrix/Vereinfachung/Bemerkung

In der Situation von Fakt gelten die Beziehungen

${}{\begin{pmatrix}\partial _{1}g_{11}\\2\partial _{1}g_{12}-\partial _{2}g_{11}\end{pmatrix}}=2{\begin{pmatrix}g_{11}&g_{12}\\g_{12}&g_{22}\end{pmatrix}}{\begin{pmatrix}\Gamma _{11}^{1}\\\Gamma _{11}^{2}\end{pmatrix}}\,,$

${}{\begin{pmatrix}2\partial _{1}g_{12}-\partial _{2}g_{11}\\\partial _{2}g_{22}\end{pmatrix}}=2{\begin{pmatrix}g_{11}&g_{12}\\g_{12}&g_{22}\end{pmatrix}}{\begin{pmatrix}\Gamma _{22}^{1}\\\Gamma _{22}^{2}\end{pmatrix}}\,$

und

${}{\begin{pmatrix}\partial _{2}g_{11}\\\partial _{1}g_{22}\end{pmatrix}}=2{\begin{pmatrix}g_{11}&g_{12}\\g_{12}&g_{22}\end{pmatrix}}{\begin{pmatrix}\Gamma _{12}^{1}\\\Gamma _{12}^{2}\end{pmatrix}}\,.$