course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
26 |
|
Paper 3, Section II, G |
2018 |
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 $U$.