I'm studying properties of flat manifolds, and I've come across the following lemma (from "Introduction to Riemannian Manifolds" by John M. Lee):
Suppose $M$ is a smooth manifold, and $\nabla$ is any connection on $M$ satisfying $$ \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z = \nabla_{[X,Y]}Z, \qquad \forall \: X, Y, Z \in \mathcal{X}(M). $$ Then given $p \in M$ and any vector $v \in T_pM$, there exists a parallel vector field $V$ on a neighborhood of $p$ such that $V_p = v$.
Lee constructs the vector field in the following way: define a coordinate cube $C_\epsilon = \{(x^1, \ldots x^n) : |x^i| < \epsilon \: \forall i\} \subset M$ centered at $p \in M$. Parallel-transport $v$ along the $x^1$-axis, then from each point $(c^1, 0, \ldots, 0)$ on the $x^1$ axis, parallel-transport the resulting vector along the coordinate curve $t \mapsto (c^1, t, 0, \ldots, 0)$ "parallel" to the $x^2$-axis. Then do the same thing to along the curve $t \mapsto (c^1, c^2, t, 0, \ldots, 0)$, etc. This eventually defines a rough vector field $V$ on the coordinate cube $C_\epsilon$.
Claim: The vector field $V$ defined like so is smooth.
What I've tried: This supposedly follows from the existence and uniqueness theorem for systems of linear differential equations:
Let $I \subset \mathbb R$ bea n open interval, and for $1 \leq j, k \leq n$, let $A_j^k : I \to \mathbb R$ be smooth functions. For all $t_0 \in I$ and every initial vector $c = (c^1, \ldots, c^n) \in \mathbb R^n$, the linear initial value problem \begin{align*} \dot V^k(t) &= A_j^k(t) V^j(t) \\ V^k(t_0) &= c^k \end{align*} has a unique smooth solution on all of $I$, and the solution depends smoothly on $(t_0,c) \in I \times \mathbb R^n$.
I can show this if $\dim M = 1$. Suppose it's true if $\dim M = n-1$. Let $N = C_{\epsilon} \cap \{x^n = 0\}$. Then $V$ is smooth on $N$. For each curve $\gamma_c(t) = (c^1, \ldots, c^{n-1}, t)$ for $c \in \mathbb R^{n-1}$, parallel-translate $V$ along $\gamma_c(t)$. Since $V$ is parallel along $\gamma_c$, $D_tV \equiv 0$, or $$ \dot V^k(t) = -\dot\gamma^i_c(t) V^j(t) \Gamma_{ij}^k(\gamma_c(t)) \quad \forall k = 1, \ldots, n, $$ where $\Gamma_{ij}^k : C_\epsilon \to \mathbb R$ are the Christoffel symbols of $\nabla$ in $C_\epsilon$-coordinates. Since $\dot\gamma_c(t) = \partial_{n}\big|_{\gamma_c(t)}$, we know $\dot\gamma_c^i \equiv \delta_n^i$. So along each curve $t \mapsto (c,t) \in C_\epsilon$, $c \in \mathbb R^{n-1}$, the coordinates of $V$ satisfy the initial value problem \begin{align*} \dot V^k(c,t) &= -V^j(c,t) \Gamma_{nj}^k\left(c^1, \ldots, c^{n-1},t\right) \\ V^k(c,0) &= V^k\big|_N(c,0) \end{align*} If we could let $A^k_j(t) = -\Gamma_{nj}^k(c,t)$, then we'd be good and the problem would be solved by induction and the existence and uniqueness theorem (since the solution depends smoothly on choice of initial condition).
My problem: I'm not sure the functions $A^k_j$ are allowed to depend on the initial condition. Changing our initial condition in this case also changes the functions $A^k_j$. Does smoothness of $V$ still follow? Alternatively, is there a reason why $\dfrac{\partial}{\partial x^i} \Gamma^k_{nj} \equiv 0$ for $1 \leq i \leq n-1$?
Oy. That's an oversight. The reference should have been to Theorem A.42 in the appendix (the fundamental theorem on flows). You need to apply that theorem to vector fields of the following form on $C_\varepsilon\times \mathbb R^n$: $$ W_k|_{(x,v)} = \frac{\partial}{\partial x^k}-v^i \Gamma^j_{ki}(x)\frac{\partial}{\partial v^j}. $$ I've added a correction to my online list.