Let $S^2(R)$ be the $2$-sphere with radius $R>0$. Then one can write in spherical coordinates the Christoffel symbols which do not depend, at least explicitly, on $R$ (see here for the computation).
My question is: selecting a small neighbourhood of the sphere, say around the north pole, and taking $R\gg1$, how can I see the effect of the curvature becoming flat in the covariant derivative, since it only depends on the Christoffel symbols? The base vectors $\partial_\theta$ and $\partial_\varphi$ becomes larger with respect to the pullback metric (by a factor $R$ I believe). Does it play any role?