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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?

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Short answer: yes. Long answer: yes indeed. Even longer answer: form the Gaussian curvature formula, you can see that if locally both $g_{ij}=0$ and $\partial g_{ij}/\partial x_k=0$ then the local curvature vanishes. – yohBS Sep 23 '12 at 12:41

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up vote 7 down vote accepted

Writing $u,v$ for parameters and $E,F,G$ for the coefficients of the fundamental form, we see on Wikipedia that when $F\equiv 0$, $$ K=-\frac{1}{2\sqrt{EG}} \left(\frac{\partial }{\partial u}\frac{G_u}{\sqrt{EG}}+\frac{\partial }{\partial v}\frac{G_v}{\sqrt{EG}} \right) $$ Which is kind of old-fashioned notation (the subscripts on the $G$s denote partial differentiation), but clear enough to conclude $K\equiv 0$; hence this cannot be the sphere.

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