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

* $R_{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}=R_{{\color {Orange}\mu }{\color {Red}\nu }{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}$
* $R_{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}=-R_{{\color {WildStrawberry}\beta }{\color {Violet}\alpha }{\color {Orange}\mu }{\color {Red}\nu }}=-R_{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }{\color {Red}\nu }{\color {Orange}\mu }}$
* ${R^{\color {Violet}\alpha }}_{<{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }>}={R^{\color {Violet}\alpha }}_{{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}+{R^{\color {Violet}\alpha }}_{{\color {Red}\nu }{\color {WildStrawberry}\beta }{\color {Orange}\mu }}+{R^{\color {Violet}\alpha }}_{{\color {Red}\nu }{\color {WildStrawberry}\beta }{\color {Orange}\mu }}=0$

Bianchi Identität

${R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{<{\color {Orange}\mu }{\color {Red}\nu };\sigma >}={R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{{\color {Orange}\mu }{\color {Red}\nu };\sigma }+{R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{\sigma {\color {Orange}\mu };{\color {Red}\nu }}+{R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{{\color {Red}\nu }\sigma ;{\color {Orange}\mu }}=0$