= Solution
{c}
The <metric tensor> in a basis $\mathbf e_a$ is $g_{ab}=\mathbf e_a\cdot\mathbf e_b$. Differentiating this <inner product> with the <product rule> and using the assumed basis-derivative relation yields
$$
\begin{aligned}
\partial_c g_{ab}
&=(\partial_c\mathbf e_a)\cdot\mathbf e_b
+\mathbf e_a\cdot(\partial_c\mathbf e_b)\\
&=\Gamma^d_{ac}\,\mathbf e_d\cdot\mathbf e_b
+\Gamma^d_{bc}\,\mathbf e_a\cdot\mathbf e_d\\
&=\boxed{\Gamma^d_{ac}g_{db}+\Gamma^d_{bc}g_{ad}.}
\end{aligned}
$$
Thus the coefficients satisfy <metric compatibility>. For a <coordinate basis> with zero <torsion tensor>, symmetrizing this relation in $a,c$, adding the corresponding two equations and subtracting the third gives
$$
\Gamma^b_{ac}=\frac12g^{bd}
(\partial_a g_{dc}+\partial_c g_{da}-\partial_d g_{ac}).
$$
They are therefore the <Christoffel symbols> of the <Levi-Civita connection>. <Metric compatibility> by itself does not establish zero <torsion tensor>; that also uses the <coordinate basis> and its commuting coordinate derivatives.
For an embedded curved surface, the stated ambient basis-derivative relation needs a qualification: the <Gauss formula> is $\partial_c\mathbf e_a=\Gamma^b_{ac}\mathbf e_b+h_{ac}\mathbf n$, where $\mathbf n$ is a <unit normal> and $h_{ac}$ is the <second fundamental form>. The normal term has zero <inner product> with each tangent basis vector, so the displayed identity for $\partial_c g_{ab}$ still holds. For a concrete counterexample to dropping that term, on a <two-sphere> one has $\partial_\theta\mathbf e_\theta=-R\mathbf n$, whereas every $\Gamma^b_{\theta\theta}$ is zero. Thus the intrinsic connection describes the tangential projection of the ambient derivative.
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