Does "$\Gamma_{k i j}=\Gamma^m{ }_{i j} g_{m k}$" mean tensor-contraction or multiplication?, i.e. $\Gamma_{k i j}=\Gamma^m{ }_{i j} \cdot g_{m k}$?

Question

Does

$\Gamma_{k i j}=\Gamma^m{ }_{i j} g_{m k}$ mean tensor-contraction or

does it mean multiplication, i.e.

$\Gamma_{k i j}=\Gamma^m{ }_{i j} \cdot g_{m k}$

An alternative definition, I cannot explain how is different is

$\Gamma_{c a b}=g_{c d} \Gamma_{a b}^d$

Does this mean contraction of tensors or does it mean multiplication?

As far as I know the Christoffel symbol does not transform like a tensor, so it's not a tensor. AFAIK, $\Gamma_{a b}^d$ is not a rank $(1,2)$ tensor, but $g_{cd}$ is a rank $(0,2)$ tensor, and $\Gamma_{c a b}=$ is not a rank $(0,3)$ tensor.

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Answer 1

As you mention the Christoffel's symbols are not tensors. Speaking of contraction in that case is i.m.o. inaccurate. The position of the suffixes in these symbols are more a convention to distinguish the type of Christoffel symbol you are dealing with. In older texts you will find the notions of "Christoffel symbol of the first kind" (with notation $\left[lm,k \right]$, equivalent to $\Gamma_{klm}$) and "Christoffel symbol of the second kind" (with notation $\left\{^k_{lm }\right\}$, equivalent to $\Gamma^k_{lm}$) with the relationship $\Gamma_{klm}=\frac{1}{2}\left(\partial_l g_{km}+ \partial_m g_{kl}-\partial_k g_{lm}\right)$ and $\Gamma^r_{st} = g^{rp}\Gamma_{pst}$ giving $g_{rq}\Gamma^r_{st} = g_{rq}g^{rp}\Gamma_{pst} = \delta^p_q\Gamma_{pst} = \Gamma_{qst}$