How to define the smooth of vector field on Riemannian manifold?

Question

$(M,g)$ is a Riemannian manifold, if $X$ is a vector field on $M$, I think for differential points $p,q\in M$, $X_p$ and $X_q$ are belong to differential space $T_pM$ and $T_qM$, I can't image how to define the smooth of vector field? I guess maybe I can define it as $g(X,X)$ is smooth on $M$, but it is define the smooth of $g$ by this way ,so I think they refer to each other.

Answer 1

A vector field at point $p$ on the differentiable manifold $M$ is smooth if in every chart $\left( {U,\varphi } \right)$ with coordinates ${x^1},...,{x^n}$ , the coefficients ${\alpha ^i}:U \to \mathbb{R}$ of the vector field in the representation ${X_p} = \sum\limits_{i = 1}^n {{\alpha ^i}\left( p \right)} \frac{\partial }{{\partial {x^i}}}{|_p}$ are differentiable functions.