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

${A^{\alpha }}_{;\mu \nu }-{A^{\alpha }}_{;\nu \mu }={R^{\alpha }}_{\beta \mu \nu }{A}^{\beta }$

Symmetrien

$R_{\alpha \beta \mu \nu }=R_{\mu \nu \alpha \beta }$