Let $S$ be a $\mathcal{C}^{\infty}$Riemannian surface. Consider $x \in S$. Can I always find two smooth vector fields $X$,$Y$ defined in a neighborhood $V$ of $x$ such that $\forall y \in V$ $(X(y),Y(y))$ form a orthonormed basis of $T_yS$ ? The point that causes me trouble is the smoothness of such vector fields. I thought maybe working with the exponential chart and taking $(\partial r, \partial \theta)$ would work, but I can't see why.
Thank you very much