# Levi-civita connection assumptions

In a lot of textbooks on relativity the Levi-Civita connection is derived like this:

$$V=V^ie_i$$
$$dV=dV^ie_i+V^ide_i$$
$$dV=\partial_jV^ie_idx^j+V^i \Gamma^{j}_{ir}e_j dx^r$$
which after relabeling indices:
$$dV=(\partial_jV^i+V^k \Gamma^{i}_{kj})e_i dx^j$$

so that the covariant derivative is defined as:

$$\nabla_j V^i=\partial_jV^i+V^k \Gamma^{i}_{kj}$$

However, the connection coefficient $$\Gamma^{i}_{kj}$$ is torsion-free by definition, as $$de_i=\Gamma^{j}_{ir}e_j dx^r$$ implies that
(1) $$\partial_r e_i=\Gamma^{j}_{ir}e_j$$.

If $$e_i=\partial_i$$ then since $$\partial_i\partial_r=\partial_r\partial_i$$ then by (1):

$$\Gamma^{j}_{ir}e_j=\Gamma^{j}_{ri}e_j$$

or that the bottom two indices are symmetric which is the torsion-free condition.

I have two questions. Is the equation $$\partial_r e_i=\Gamma^{j}_{ir}e_j$$ true in general for any connection? And also, where did the torsion-free assumption enter into the derivation above?