I am reading the book Information geometry by "Shun-Ichi Amari",and he has written one line where I am confused
"If $S$ has a Riemannian metric $g$ then we may define $n^3$ additional functions $\Gamma_{ij,k}$ in the following way:"
$$\Gamma_{ij,k}=\langle \nabla_{\partial_{i}}\partial_{j},\partial_{k}\rangle=\Gamma_{ij}^{h}g_{hk} \tag{1}$$
The quantities $\{\Gamma_{ij,k}\}$, like $\Gamma_{ij}^{k}$ may be considered as a different components expression of the same $\nabla.$
What I know is that $\nabla_{\partial_{i}}\partial_{j}=\Gamma_{ij}^{k}\partial_{k}.$ I think we are getting $(1)$ from here only but not sure.Can someone help me that how we got $(2)$,Thanks.
$$ \nabla_{\partial_i}\partial_j=\Gamma_{ij}^l\partial_l $$ implies $$ \langle\nabla_{\partial_i}\partial_j,\partial_k\rangle= \Gamma_{ij}^l\langle\partial_l,\partial_k\rangle=\Gamma_{ij}^lg_{lk} $$