It seems as though in writing down all formulae of Riemannian geometry in local coördinates, the Einstein summation convention works mysteriously, and I don't have a clue as to the reason for that. Indeed, if we write all vectors in local coordinates, e.g.
$$ X = \lambda^i \frac{\partial}{\partial x_i} $$
and run computations, it never once occurs that something cannot be written in the Einstein convention. But what is the reason for that?