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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

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The exponential chart is the answer, but why use polar coordinates? Just use the usual orthonormal coordinates. – Zhen Lin Feb 25 '12 at 17:48

Gram-Schmidt: locally you do have a smooth frame $X=\partial_1, Y=\partial_2$ from a chart. you can surly define $X_1:=X/\vert|X\|$, a smooth vector field. Now define $$ Y_0:=Y-g(Y,X_1)X_1 \quad Y_1:=Y_0/\vert|Y_0\| $$ Now $X_1,Y_1$ are smooth and orthonormed.

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