Covariant derivative of a constant vector field

Question

I am trying to figure out that if $X$ is locally constant, i.e. in some coordinate system, the coefficient function is a constant function, then whether the covariant derivative of $X$ is 0. In $\mathbb R^n$ I guess this is true. But what about the general case, how can I derive this from the properties of covariant derivative?

Answer 1

This is certainly not true - in fact, for any non-vanishing vector field $X$ there is a coordinate system $x^i$ in which $X = \partial/\partial x^1$. Remember the coordinate formula for the covariant derivative is $$\nabla_i X^j = \partial_i X^j + \Gamma_{ik}^j X^k,$$ so the equivalence of $X$ being parallel (zero covariant derivative) and having constant components holds only when the Christoffel symbols vanish. This is true only in flat spaces, and even then only in particular coordinates.