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Curvature, Riemann and Ricci tensors, Ricci scalar

Covariant derivative in general is not commutative, ${V^{\alpha}}_{; \nu ; \mu} - {V^{\alpha}}_{; \mu ; \nu} \ne 0$. Namely
\begin{displaymath}
{V^{\alpha}}_{; \nu ; \mu} - {V^{\alpha}}_{; \mu ; \nu} \equiv
{R^{\alpha}}_{\gamma\mu\nu} V^{\gamma}
\end{displaymath} (16)

defines Riemann tensor ${R^{\alpha}}_{\gamma\mu\nu} $ which gives invariant measure of the curvature of the space. The space is flat if ${R^{\alpha}}_{\gamma\mu\nu} = 0 $.
Riemann tensor ${R^{\alpha}}_{\gamma\mu\nu} $

\begin{displaymath}
{R^{\alpha}}_{\beta\mu\nu} = \Gamma^{\alpha}_{\beta\nu , \mu...
...a\nu}
- \Gamma^{\alpha}_{\gamma\nu} \Gamma^{\gamma}_{\beta\mu}
\end{displaymath} (17)

Ricci tensor $R_{\alpha\beta}$
is the contraction of the Riemann tensor
\begin{displaymath}
R_{\alpha\beta} \equiv {R^{\gamma}}_{\alpha\gamma\beta}
\end{displaymath} (18)

Ricci scalar $R$
is the further contraction
\begin{displaymath}
R \equiv g^{\alpha\beta} R_{\alpha\beta} = {R^\alpha}_{\alpha}
\end{displaymath} (19)



Dmitri Pogosyan 2006-09-28