This is probably a dumb question, but I'm sure someone here can set me straight.
In do Carmo's "Riemannian Geometry" the covariant derivative is computed in local coordinates. He then goes on to talk about compatible metrics. In particular, in the proof that an affine connection is compatible with the metric, then we can differentiate the inner product along a differentiable curve $c$, and recover a product rule: $\frac{d}{dt} \langle V, W \rangle= \langle \frac{DV}{dt}, W \rangle + \langle V, \frac{DW}{dt} \rangle$.
The proof goes like this:
Choose an orthornomal basis $\{ P_1 (t'), ..., P_n (t') \}$ of the tangent space at $t'$ along the curve $c$. Using a past proposition, we can use this to create a basis for the tangent space along every point of the curve using parallel transport. Since the connection is compatible with the metric, this basis remains orthonormal (because the inner product of any pair is constant, and was $0$ at the point).
So far so good. Here's where I get lost.
We can write $V = \sum v^i P_i$, and $W = \sum w^i P_i$, and differentiate. Do Carmo claims that $\frac{DV}{dt} = \sum \frac{dv^i}{dt} P_i$, and similarly for $W$. Where did the Christoffel symbols go?
Some comparison to other results seems to suggest that they vanish because the coordinate system is orthonormal. In particular, it can be shown for surfaces that the Christoffel symbols vanish when using normal coordinates from the exponential map. But none of these notions have been defined for Riemannian manifolds. Somehow I want to be able to figure out solely from the properties of an affine connection (or whatever other foundational notions are required) why the Christoffel symbols poofed. Nothing about curvature or geodesics has been introduced, so I should like to avoid those notions if I can help it.
$DP_i/dt = 0$ because the vector fields $P_i$ are parallel along the curve. Just that simple. (You can't have an orthonormal coordinate system unless the manifold is flat. This is much weaker — just along a curve.)