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}}