Trying to understand the motivation of the curvature tensor in Lee's book about Riemannian manifolds

Question

I am trying to understand the following passage from Lee's book on Riemannian manifolds. Unfortunately there are a few things I am not sure about.

Questions:

1. Why do we have $\nabla_{\partial_{1}}Z=0$, if $x^{2}=0$. I tried to use

as well as

2. Why is the following statement true? My guess is because of ODEs.

Thank you very much in advance.

Answer 1

1. $Z$ is constructed by first parallel transporting $z \in T_pM$ first along the $x^1$-axis, i.e., along the coordinate curve $t \mapsto (t, 0)$. The tangent vector to this curve is $\partial_1$, so this just means that $\nabla_{\partial_1}Z = 0$ when $x^2 = 0$.
2. This is by uniqueness of parallel transport, which is (as you suggest) a consequence of the uniqueness of solutions to linear ODEs. See Theorem 4.31 and Theorem 4.32 in the same book.