R^2/Offene Menge/Riemannsche Metrik/Levi-Civita-Zusammenhang/Krümmungsoperator/Berechnung/Aufgabe/Lösung

Aufgrund der verschiedenen Identitäten aus Fakt sind allenfalls die Christoffelsymbole

$R_{112}^{1},R_{112}^{2},R_{122}^{1},R_{122}^{2},$

relevant. Es ist gemäß Fakt

${}R_{112}^{1}={\partial _{1}}\Gamma _{12}^{1}-{\partial _{1}}\Gamma _{12}^{1}+\Gamma _{12}^{1}\Gamma _{11}^{1}+\Gamma _{12}^{2}\Gamma _{12}^{1}-\Gamma _{12}^{1}\Gamma _{11}^{1}-\Gamma _{12}^{2}\Gamma _{12}^{1}=0\,,$

${}R_{112}^{2}={\partial _{1}}\Gamma _{12}^{2}-{\partial _{1}}\Gamma _{12}^{2}+\Gamma _{12}^{1}\Gamma _{11}^{2}+\Gamma _{12}^{2}\Gamma _{12}^{2}-\Gamma _{12}^{1}\Gamma _{11}^{2}-\Gamma _{12}^{2}\Gamma _{12}^{2}=0\,,$

${}R_{121}^{1}=-{\partial _{1}}\Gamma _{12}^{1}-{\partial _{2}}\Gamma _{11}^{1}-\Gamma _{12}^{1}\Gamma _{11}^{1}-\Gamma _{12}^{2}\Gamma _{12}^{1}+\Gamma _{11}^{1}\Gamma _{12}^{1}-\Gamma _{11}^{2}\Gamma _{22}^{1}\,,$

${}R_{121}^{2}={\partial _{1}}\Gamma _{12}^{2}-{\partial _{2}}\Gamma _{11}^{2}+\Gamma _{12}^{1}\Gamma _{11}^{2}+\Gamma _{12}^{2}\Gamma _{12}^{2}-\Gamma _{11}^{1}\Gamma _{12}^{2}-\Gamma _{11}^{2}\Gamma _{22}^{2}\,.$

${}R_{122}^{1}={\partial _{1}}\Gamma _{22}^{1}-{\partial _{2}}\Gamma _{22}^{1}+\Gamma _{22}^{1}\Gamma _{11}^{1}+\Gamma _{22}^{2}\Gamma _{12}^{1}-\Gamma _{12}^{1}\Gamma _{12}^{1}-\Gamma _{12}^{2}\Gamma _{22}^{1}\,,$

${}R_{122}^{2}={\partial _{1}}\Gamma _{22}^{2}-{\partial _{2}}\Gamma _{22}^{2}+\Gamma _{22}^{1}\Gamma _{11}^{2}-\Gamma _{12}^{1}\Gamma _{12}^{2}\,.$

Zur gelösten Aufgabe