In what coordinates do the Christoffel symbols of this flat connection vanish? Let $M$ be a submanifold of a pseudo-Riemannian manifold $N$ such that the Levi-Civita connection of $N$ is flat. For simplicity, since this seems to already include the general case, let us restrict in this question to the case of $M = S^{d - 1}$ embedded the usual way in $\mathbb R^d$. For $d \geq 3$, the Levi-Civita connection on $TM$ is obviously not flat, but we can embed $TM$ in the bundle $V := TM \oplus NM = M \times \mathbb R^d$. The natural connection $D$ on $V$ is defined by $D_v X = \nabla_v X$, i.e. it is the restriction of the Levi-Civita connection on $\mathbb R^d$. Thus for vector fields $v, w$ tangent to $M$ we have
$$[D_v, D_w] = [\nabla_v, \nabla_w] = [v, w]$$
i.e. $D$ is flat. So there exists a coordinate frame on $M$ such that all the Christoffel symbols of $M$ vanish.
However, it's not clear to me what that coordinate frame should be. Some candidates are the coordinates you get by thinking of the half-spheres as graphs (i.e. just forget about the $d$ coordinate and let $x^i$ be the $i$th coordinate on $\mathbb R^d$, $i = 1, \dots, d - 1$), and the projection of any set of $d-1$ basis vectors of $\mathbb R^d$ onto the tangent space.
When I try to write out the connection in such coordinates and go through the manipulations, I mostly manage to give myself a headache rather than get a computation that seems to "go somewhere" and tell me what the Christoffel symbols are. But it seems like there should be some general theorem which tells me what coordinate frame to use to kill the Christoffel symbols, allowing me to avoid such tedious computations. Is there such a theorem?
 A: It might be useful to review the definition of the Christoffel symbols of a connection on an arbitrary vector bundle. If $M$ is an $n$-dimensional manifold, $V\to M$ a rank $r$ vector bundle and $D$ is a connection on $V$, the connection can be locally described by a $1$-form.
Fix a local frame $\mathbf{s}=(s_1,\dots,s_r)$ for $V\to M$, i.e. a basis of local sections of $V$ over some open set $U\subseteq M$. Differentiating the $s_\alpha$ we obtain $$Ds_\alpha=:A^\beta_\alpha\otimes s_\beta$$
for a matrix $A=\left(A^\beta_\alpha\right)_{1\leq\alpha,\beta\leq r}$ of $1$-forms on $U$. Fixing a system of coordinates $\mathbf{x}=(x^1,\dots,x^n)$ for $M$ defined on $U$, each of the $1$-forms $A^\beta_\alpha$ can be written as $$A^\beta_\alpha=A^\beta_{\alpha\,a}\mathrm{d}x^a.$$ Notice that we are using greek indices $\alpha,\beta,\dots$ for the sections of $V$, and latin indices $a,b,\dots$ for the coordinates on $M$.
What usually happens in the Riemannian context is that you consider a particular connection on $V=TM$, and after having fixed a system of local coordinates $\mathbf{x}$ for $M$ you get the local frame of $M$ defined by $\partial_{x^1},\dots,\partial_{x^n}$, so the connections can locally be written as $$\nabla\partial_{x^a}=A^b_{ac}\mathrm{d}x^c\otimes\partial_{x^b}.$$ In this context we are allowed, to some extent, to confuse the local system of coordinates with the local frame for $V$. What we usually call Christoffel symbols are just the components $A^b_{ac}$.
Now, after this long introduction, let's get to the question. My point is that the right question is not "what are the flat coordinates on $M$", but rather "what is the flat frame for $V$". The first question really makes sense only for a connection on $TM$, while the second one can be asked of any flat connection on any vector bundle. In your case, we are considering a trivial vector bundle $M\times\mathbb{R}^d\to M$, and as such we have a global frame $s_1,\dots,s_d$ defined by $$s_\alpha(p)=(p,e_d),$$ where $e_1,\dots,e_d$ is a basis for the vector space $\mathbb{R}^d$. It seems to me that for the connection you have defined, this is the flat frame for $V$. Let me know if you find this useful!
A: What you're looking for are perhaps geodesic normal coordinates. For example, reference here: Wikipedia.
