Given a parametrized surface $\phi(x,y) = (x,y,x^2-y^2)$ and an orthonormal frame for the tangent bundle $$ v_1(x,y) = \frac{1}{\sqrt{1+4x^2}}\Big(-y,x,0\Big) \\ v_2(x,y) = \frac{1}{\sqrt{(x^2+y^2)(x^2-y^2)^2 + (x^2+y^2)^2}}\Big(x(x^2-y^2),y(x^2-y^2),-x^2-y^2\Big) $$ I would like to be able to calculate $\nabla_{v_i}$, so that I can calculate the curvature (https://en.wikipedia.org/wiki/Riemann_curvature_tensor) on the frame. My understanding is that it should be possible to relate a tangent vector to a vector field (Why is every tangent vector part of a vector field?), possibly not uniquely.
Can anyone please indicate how I would go about this? From the linked question, it seems like I need to choose some $\varphi_i$ such that $\frac{\partial}{\partial \varphi_i}\mid_{x,y} = v_i(x,y)$. Is this the best approach? Since $v_1$ and $v_2$ are orthonormal, does it imply anything about the corresponding commutation relations for the $\frac{\partial}{\partial \varphi_i}$?