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I'm trying to solve this exercise from Kühnel.

Let $f: [0, A] \times [0, B] \rightarrow \mathbb{R}^3$ be a parametrized surface element. Show that the following conditions (1) and (2) are equivalent:

  1. For each rectangle $R = [u_1, u_1 + a] \times [u_2, u_2 + b] \subset U$, the opposite sides of $f(R)$ are of equal length.
  2. One has $\frac{\partial g_{11}}{\partial u_2} = \frac{\partial g_{22}}{\partial u_1} = 0$ in all of $U$. ($g_{ij}$ are the matrix entries of the first fundamental form)

I've solved (2) $\implies$ (1) and after much trying, I don't know how to prove the other implication.

Kühnel further asks to prove that under these conditions, there is a parameter transformation $\varphi: U \rightarrow \tilde{U}$ such that for $\tilde{f} = f \circ \varphi^{-1}$ the first fundamental form can be written as:

\begin{pmatrix} 1 & \cos \vartheta \\ \cos \vartheta & 1 \end{pmatrix}

and I don't see how this follows from setting $\varphi(u_1, u_2) = \left( \int \sqrt{g_{11}} du_1, \int \sqrt{g_{22}} du_2 \right)$.

1 Answers 1

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HINTS: (1) Suppose $\frac{\partial g_{11}}{\partial u_2}>0$ on some open set containing $[a,a+h]\times [b,b+k]$. Show that the length of $f\big([a,a+h]\times\{b+k\}\big)$ is greater than the length of $f\big([a,a+h]\times\{b\}\big)$.

(2) If you set $(u'_1,u'_2) = \phi(u_1,u_2)$, what is the length of $\partial\tilde f/\partial u'_1$?

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    If $\alpha (t) = (t, b+k)$ and $\beta (t) = (t, b)$ are the sides of the rectangle, their lengths on the curve are $\int_a^{a+h} \sqrt{g_{11}(\alpha (t))} dt$, and $\int_a^{a+h} \sqrt{g_{11}(\beta (t))} dt$, respectively. I don't see how one can use the fact that $\frac{\partial g_{11}}{\partial u_2} > 0$ to show that they are different. – 2017-02-13
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    Remember what a positive derivative tells you! – 2017-02-13
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    That the function is increasing... So? – 2017-02-13
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    So compare those two integrals! – 2017-02-13
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    What did you just tell me about $g_{11}(t,b+k)$ and $g_{11}(t,b)$? – 2017-02-13
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    $g_{11}(t, b + k)$ must be greater than $g_{11}(t,b)$ for all $t$. Ok, thank you. – 2017-02-13