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

* $R_{\alpha \beta \mu \nu }=R_{\mu \nu \alpha \beta }$
* $R_{\alpha \beta \mu \nu }=-R_{\beta \alpha \mu \nu }=-R_{\alpha \beta \nu \mu }$
* ${R^{\alpha }}_{<\beta \mu \nu >}={R^{\alpha }}_{\beta \mu \nu }+{R^{\alpha }}_{\nu \beta \mu }+{R^{\alpha }}_{\nu \beta \mu }=0$

Bianchi Identität

${R^{\alpha \beta }}_{<\mu \nu ;\sigma >}={R^{\alpha \beta }}_{\mu \nu ;\sigma }+{R^{\alpha \beta }}_{\sigma \mu ;\nu }+{R^{\alpha \beta }}_{\nu \sigma ;\mu }=0$