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Covariant derivative in general is not commutative,
. Namely
![\begin{displaymath}
{V^{\alpha}}_{; \nu ; \mu} - {V^{\alpha}}_{; \mu ; \nu} \equiv
{R^{\alpha}}_{\gamma\mu\nu} V^{\gamma}
\end{displaymath}](img62.png) |
(26) |
defines Riemann tensor
which gives invariant measure of the curvature of the space.
The space is flat if
.
- Riemann tensor
![${R^{\alpha}}_{\gamma\mu\nu} $](img63.png)
-
![\begin{displaymath}
{R^{\alpha}}_{\beta\mu\nu} = \Gamma^{\alpha}_{\beta\nu , \mu...
...a\nu}
- \Gamma^{\alpha}_{\gamma\nu} \Gamma^{\gamma}_{\beta\mu}
\end{displaymath}](img65.png) |
(27) |
- Ricci tensor
![$R_{\alpha\beta}$](img66.png)
- is the contraction of the Riemann tensor
![\begin{displaymath}
R_{\alpha\beta} \equiv {R^{\gamma}}_{\alpha\gamma\beta}
\end{displaymath}](img67.png) |
(28) |
- Ricci scalar
![$R$](img68.png)
- is the further contraction
![\begin{displaymath}
R \equiv g^{\alpha\beta} R_{\alpha\beta} = {R^\alpha}_{\alpha}
\end{displaymath}](img69.png) |
(29) |
Dmitri Pogosyan
2009-10-23