Paper 3, Section II, G

Let $U$ be an open subset of the plane $\mathbb{R}^{2}$, and let $\sigma: U \rightarrow S$ be a smooth parametrization of a surface $S$. A coordinate curve is an arc either of the form

$\alpha_{v_{0}}(t)=\sigma\left(t, v_{0}\right)$

for some constant $v_{0}$ and $t \in\left[u_{1}, u_{2}\right]$, or of the form

$\beta_{u_{0}}(t)=\sigma\left(u_{0}, t\right)$

for some constant $u_{0}$ and $t \in\left[v_{1}, v_{2}\right]$. A coordinate rectangle is a rectangle in $S$ whose sides are coordinate curves.

Prove that all coordinate rectangles in $S$ have opposite sides of the same length if and only if $\frac{\partial E}{\partial v}=\frac{\partial G}{\partial u}=0$ at all points of $S$, where $E$ and $G$ are the usual components of the first fundamental form, and $(u, v)$ are coordinates in $U$.

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