We now outline the derivation of an explicit formula for the connection 1-forms, thus also proving that, given $d\rr$, there is a unique Levi-Civita connection. We emphasize, however, that this formula is rarely the most efficient way to actually compute the connection 1-forms.

Introduce the components of the connection 1-forms by writing \begin{equation} \omega_{ij} = \Gamma_{ijk}\,\sigma^k \end{equation} The components $\Gamma_{ijk}$ are known as Christoffel symbols of the first kind. Metric compatibility implies \begin{equation} 0 = \omega_{ji} + \omega_{ij} = \left( \Gamma_{ijk}+\Gamma_{jik} \right) \sigma^k \label{mcomp2} \end{equation} so that \begin{equation} \Gamma_{ijk} + \Gamma_{jik} = 0 \end{equation} The torsion-free condition further implies that \begin{equation} 0 = d\sigma_i + \omega_{ij}\wedge\sigma^j = d\sigma_i + \Gamma_{ijk}\,\sigma^k\wedge\sigma^j \label{tfree2} \end{equation} where we have introduced \begin{equation} \sigma_p = g_{pj}\sigma^j \end{equation} Note that \begin{equation} g(\sigma^i,\sigma_p) = \delta^i{}_p \end{equation} since the sign in $g(\sigma^i,\sigma^j)$ cancels the one in the definition of $\sigma_p$. Thus, we also have \begin{equation} g(\sigma^i\wedge\sigma^j,\sigma_p\wedge\sigma_q) = \delta^i{}_p \delta^j{}_q - \delta^i{}_q \delta^j{}_p \end{equation} using the definition of $g$ on 2-forms, and our new “up and down” index notation to keep track of the signs. Thus, \begin{equation} g(d\sigma_i,\sigma_p\wedge\sigma_q) = - \Gamma_{ijk} \,g(\sigma^k\wedge\sigma^j,\sigma_p\wedge\sigma_q) = \Gamma_{ipq} - \Gamma_{iqp} \end{equation}

We can now find a formula for $\Gamma_{ijk}$, and hence for $\omega_{ij}$, by writing \begin{align} 2\Gamma_{ijk} &= \Gamma_{ijk} + \Gamma_{ijk} \nonumber\\ &= \Gamma_{ijk} - \Gamma_{jik} \nonumber\\ &= \Gamma_{ijk} + \left( - \Gamma_{ikj} + \Gamma_{ikj} \right) + \left( -\Gamma_{jki} + \Gamma_{jki} \right) - \Gamma_{jik} \nonumber\\ &= \left( \Gamma_{ijk} - \Gamma_{ikj} \right) + \left( -\Gamma_{kij} + \Gamma_{kji} \right) + \left( \Gamma_{jki} - \Gamma_{jik} \right) \nonumber\\ &= g(d\sigma_i,\sigma_j\wedge\sigma_k) - g(d\sigma_k,\sigma_i\wedge\sigma_j) + g(d\sigma_j,\sigma_k\wedge\sigma_i) \label{koszul} \end{align}

The construction above will still work even if the left-hand side of ($\ref{mcomp2}$) and ($\ref{tfree2}$) are nonzero but known; the result is an expression for $\Gamma_{ijk}$ which will contain those nonzero terms, and is known as the Koszul formula. Thus, the “torsion” and “non-metricity” completely determine the connection, regardless of whether they are zero; the result for Levi-Civita connections is a special case.