Riemannscher Krümmungstensor

${R^{\color {Violet}\alpha }}_{{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}=\Gamma _{{\color {WildStrawberry}\beta }{\color {Orange}\mu },\color {Red}\nu }^{\color {Violet}\alpha }-\Gamma _{{\color {WildStrawberry}\beta }{\color {Red}\nu },{\color {Orange}\mu }}^{\color {Violet}\alpha }+\Gamma _{{\color {WildStrawberry}\beta }{\color {Orange}\mu }}^{\sigma }\Gamma _{\sigma {\color {Red}\nu }}^{\color {Violet}\alpha }-\Gamma _{{\color {WildStrawberry}\beta }{\color {Red}\nu }}^{\sigma }\Gamma _{\sigma {\color {Orange}\mu }}^{\color {Violet}\alpha }$

Herleitung

${A^{\color {Violet}\alpha }}_{;{\color {Orange}\mu }{\color {Red}\nu }}-{A^{\color {Violet}\alpha }}_{;{\color {Red}\nu }{\color {Orange}\mu }}={R^{\color {Violet}\alpha }}_{{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}{A^{\color {WildStrawberry}\beta }}$

Symmetrien

$R_{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}=R_{{\color {Orange}\mu }{\color {Red}\nu }{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}$