I am wondering in what sense the formula of contraction of indices for the connection coefficients of the Levi-Civita connection (i.e. the Christoffel symbols) is valid for other connections.
\begin{equation} \tag{1} \Gamma ^{i}{}_{ki}={\frac {\partial \ln {\sqrt {|g|}}}{\partial x^{k}}} \end{equation}
My questions:
Is (1) specifically valid for the Levi-Civita connection or does it also hold for other (metric) connections, maybe if they have certain properties like being torsion free? (I learned that torsion-free is equivalent to the connection coefficients being symmetric in the lower indices, so I suspect it is crucial for (1) to hold.)
Is there a more general contraction law for other connections (maybe with certain properties), e.g. (1) extended by some additional term?