Affine connection (Christoffel symbols)

Affine connection $\Gamma^{\alpha}_{\beta\gamma}$ describes relation between vectors at two neighbouring points.

$$\delta V^{\alpha} = - \Gamma^{\alpha}_{\beta\gamma} V^{\beta} dx^{\gamma}$$ (20)

Covariant derivatives

We denote ordinary derivatives with comma and the covariant ones with semicolon

$$S_{;\mu} = S_{,\mu}   -    for scalars derivatives are equal.$$ (21)

$${V^{\alpha}}_{;\mu} = {V^{\alpha}}_{,\mu} + \Gamma^{\alpha}_{\mu\gamma} V^{\gamma}$$ (22)

$$V_{\alpha;\mu} = V_{\alpha,\mu} - \Gamma^{\gamma}_{\mu\alpha} V_{\gamma}$$ (23)

$${T^{\beta}}_{\alpha;\mu} = {T^{\beta}}_{\alpha,\mu} - \Gamma^{\gamma}_{\mu\alpha} {T^{\beta}}_{\gamma} + \Gamma^{\beta}_{\mu\gamma} {T^{\gamma}}_{\alpha}$$ (24)

Relation between $\Gamma^{\alpha}_{\beta\gamma}$ and $g_{\alpha\beta}$:

In GR we use affine connection which is related to the first derivatives of the metric tensor by the requirement that $g_{\alpha\beta ; \mu} = 0$ and restriction that connection is symmetric $\Gamma^{\alpha}_{\beta\gamma} = \Gamma^{\alpha}_{\gamma\beta}$. Then

$$\Gamma^{\alpha}_{\beta\gamma} = \frac{1}{2} g^{\alpha\sigma}... ... + g_{\sigma\gamma , \beta} - g_{\beta\gamma , \sigma} \right)$$ (25)