Reference Request on a Necessary and Sufficient Condition for Isothermality

I am getting acquainted with the fundamentals of differential geometry for the sake of a problem I have been thinking about.

In

Green, G. M. "Some Geometric Characterization of Isothermal Nets on a Curved Surface." Transactions of the American Mathematical Society, Vol. 18, No. 4, (1917) pp. 480-488,

the author makes the following preamble:

'If a surface $S$, whose equations are

$$ \begin{equation} x=x(u,v), \hspace{2mm} y=y(u,v), \hspace{2mm}z=z(u,v), \end{equation} $$ be referred to an orthogonal net of parameter curves, its first fundamental form is $$ \begin{equation} ds^2=Edu^2+Gdv^2, \end{equation} $$ where $E$ and $G$ are functions of $u$ and $v$. This orthogonal net is called isothermal, if by a proper transformation $\bar{u}=\phi(u),\bar{v}=\psi(v)$ of the parameters the first fundamental form may be reduced to $$\begin{equation} ds^2=\bar{\lambda}(\bar{u},\bar{v})(d\bar{u}^2+d\bar{v}^2). \end{equation}$$

A necessary and sufficient condition that such a reduction may be effected upon the form as first written is, that the equation

$$ \begin{equation} \frac{\partial^2}{\partial u \partial v} log \Big(\frac{E}{G} \Big)=0 \end{equation} $$ be satisfied identically.'

I would like to know

1. the name of this result (if any);
2. where it originated;
3. whether it has variants pertaining to the classification of surfaces where the middle term of the first fundamental form does not vanish.

Textbooks that would answer my questions are welcome.

Thanks for your attention!

Solution 1:

I've never seen this result before, but I'll show you a proof. The point of isothermal coordinates is to give a conformal flat structure, i.e., a metric in which angles are equivalent to Euclidean angles in the plane. Here we ordinarily use the standard coordinates on the plane and so we want $F=0$ for sure.

OK, let's consider the partial differential equation $$\frac{\partial^2 f}{\partial u\partial v} = 0$$ on a convex open subset of the $(u,v)$-plane. It is straightforward to see that $f(u,v)=g(u)+h(v)$ for some functions $g$ and $h$. Since we will be interested in the case $f=\log F$, we observe that we want to write $\log F = \log G-\log H=\log(G/H)$ with $G,H>0$; locally we can obtain this by shifting by appropriate constants if necessary.

When we write $u=\phi(\bar u)$, $v=\psi(\bar v)$ the metric becomes $\bar E\,d\bar u^2 + \bar G\,d\bar v^2$ with $\bar E = (\phi'(u))^2E$ and $\bar G=(\psi'(v))^2G$. We see that $(\bar u,\bar v)$ give isothermal coordinates if and only if $$(\phi'(u))^2E = \bar E = \bar G = (\psi'(v))^2G \iff \frac EG = \frac{\psi'(v)^2}{\phi'(u)^2}.$$ By our earlier remarks, $\log(E/G)$ is the sum of a function of $u$ and a function of $v$ if and only if $\dfrac{\partial^2}{\partial u\partial v}\log(E/G) = 0$.

REMARK: It is a long-standing theorem that every (sufficiently smooth) surface admits isothermal coordinates. A relatively accessible reference is this: "An Elementary Proof of the Existence of Isothermal Parameters on a Surface," S-S. Chern, Proc. Amer. Math Soc. 6 (1955), 771-782.