Let $f:V \to S^{2} \subset \mathbb{R}^{3}$ be a surface whose image is an open subset of the usual unit sphere $S^2$. Prove that there is NOT a change of variables $\phi: U \to V$ such that in the new variables the components of the first fundamental form are $g_{ij}=\delta_{ij}$ where $g_{ij}= \langle f_{v^{1}} ,f_{v^{2}} \rangle$ and $\delta _{ij}$ is the Kronecker delta.
Note: There is a result states that the Gauss curvature of $f$ at $v$ is zero if and only if there exists local coordinates in which the matrix rep of the first fundamental form of $f$ at $v$ is the identity matrix under the coordinates (for instance one may take the Fermi local coordinates).
Based on this result, is it true that if there is a global change of variables which makes $g_{ij}=\delta_{ij}$ hold then this change of varibles can also be considered as locally at each point on the surface, thus by above result the Gauss curvature of each point on the surface is zero?