Riemannscher Krümmungstensor: Unterschied zwischen den Versionen
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*>SchuBot Einrückungen Mathematik |
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(5 dazwischenliegende Versionen von einem anderen Benutzer werden nicht angezeigt) | |||
Zeile 15: | Zeile 15: | ||
== Herleitung == | == Herleitung == | ||
<math> | :<math> | ||
{A^{\color{Violet}\alpha}}_{;{\color{Orange}\mu} {\color{Red}\nu}} | {A^{\color{Violet}\alpha}}_{;{\color{Orange}\mu} {\color{Red}\nu}} | ||
- | - | ||
Zeile 21: | Zeile 21: | ||
= | = | ||
{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}} | ||
{A^{\color{ | {A^{\color{WildStrawberry}\beta}} | ||
</math> | |||
== Symmetrien == | |||
* <math> | |||
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} } | |||
</math> | |||
* <math> | |||
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}} | |||
</math> | |||
* <math> | |||
{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 | |||
</math> | |||
=== Bianchi Identität === | |||
:<math>{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 | |||
</math> | </math> | ||
[[Kategorie:Tensor 4. Stufe]] | [[Kategorie:Tensor 4. Stufe]] |