-
$$
R_{iklm}=g_{in}R^n_{klm}
=
\frac{1}{2}\left(\frac{\partial^2 g_{im}}{\partial x^k \partial x^l}
+\frac{\partial^2 g_{kl}}{\partial x^i \partial x^m}
-\frac{\partial^2 g_{il}}{\partial x^k \partial x^m}
-\frac{\partial^2 g_{km}}{\partial x^i \partial x^l}
\right)$$
$$
+g_{np}\left(\Gamma^n_{kl}\Gamma^p_{im}-\Gamma^n_{km}\Gamma^p_{il}\right)
$$
-
$$
R_{iklm}=-R_{kilm}=-R_{ikml}
$$
-
$$
R_{iklm}=R_{lmik}
$$
-
$$
R_{iklm}+R_{imkl}+R_{ilmk}=0
$$
- Bianchi-azonosság:
$$
R^n_{ikl;m}+R^n_{imk;l}+R^n_{ilm;k}=0
$$
Bizonyítás lokálisan geodetikus rendszerben:
$$
R^n_{ikl;m}=\frac{\partial^2 R^n_{ikl}}{\partial x^m}
=\frac{\partial^2 \Gamma^n_{il}}{\partial x^m \partial x^k}
-\frac{\partial^2 \Gamma^n_{ik}}{\partial x^m \partial x^l}
$$
- Ricci-tenzor:
$$
R_{ik}=g^{lm}R_{limk}=R^m_{imk}
$$
$$
R_{ik}=R_{ki}
$$
- Invariáns görbület:
$$
R=g^{ik}R_{ik}
$$
- Kétdimenziós eset:
$$
R=\frac{2P_{1212}}{\gamma}
$$
$$
\frac{P}{2}=K=\frac{1}{\rho_1\rho_2}
$$
(levezetés)
