On a 2-Riemannian manifold $M$ one can try to extend a vector field in the following manner. Given $p \in M$, $V \in T_pM$, choose a coordinate neighborhood $(x^1,x^2)$ around $p$. Can first parallel transport $V$ along the $x^1$ axis, then parallel transport the resulting vectors along the $x^2$ lines. This would be an extension of the vector field, denoted by $Z$. Then I was told that one could argue that on a flat manifold $\nabla_{\partial_1}Z = 0$.
Two questions:
1). Why is the resulting $Z$ smooth? Generally, the method used to prove its smoothness is to write it in terms of bases, but I find it hard to do so here.
2).Why is it good to have $\nabla_{\partial_1}Z = 0$ given our initial mission is just to extend a vector?