I am trying to prove the following equivalence for a parametrized surface $f: U \to \mathbb{R}^3 $ with $U = (0,A) \times (0,B) $ :
- For every Rectangle $R$ in $U$, the opposite sides of $f(R)$ have the same length.
- $\partial_{u_1}g_{22} = \partial_{u_2}g_{11} \equiv 0$ on $U$.
I proved ($2. \implies 1.$). But for the converse, i have an argument, but i am not sure how formaly acceptable it is. Here it goes : (Note: I will just write $g$ for $g_{11}$ or $g_{22}$)
Assuming the set on which the partial is non zero has non zero measure, using that g is continuous on a connected set of $\mathbb{R^{2}}$, you can find an $\epsilon$ and a $\delta$, such that $\mu(\{g > \epsilon +\delta\})$ and $\mu(\{g < \epsilon - \delta\})$ both have non zero measure and you can parametrize a rectangle such that one side goes through $\{g > \epsilon +\delta\}$ and the opposite side through $\{g < \epsilon - \delta\}$.
If this is ok, then i have a bound on $g$, and since the $L(\gamma) = \int_I \sqrt{g} \ dt$, evaluating and bounding the integral for both sides yields the result.
I am not confident because it feels strange to mix some "pure" measure theory into this proof. As mixing a parameter and a measure feels akward.
I think i can utilize a similar argument, using an open set around a point where the partial is non-vanishing (thus not using measure theory), but i am curious about what people here think of that first argument.