Riemannscher Krümmungstensor

${R^{\alpha }}_{\beta \mu \nu }={\mathrm{\Gamma }}_{\beta \mu ,\nu }^{\alpha }-{\mathrm{\Gamma }}_{\beta \nu ,\mu }^{\alpha }+{\mathrm{\Gamma }}_{\beta \mu }^{\sigma }{\mathrm{\Gamma }}_{\sigma \nu }^{\alpha }-{\mathrm{\Gamma }}_{\beta \nu }^{\sigma }{\mathrm{\Gamma }}_{\sigma \mu }^{\alpha }$

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}}