Unit speed reparametrizations of proper patches

Question

Suppose that $x:D\to M\subset\mathbb{R}^3$ is a proper patch, where $D$ is some open set in $\mathbb{R}^2$. The first fundamental form involves the real-valued functions,
$E=x_u.x_u$,
$F=x_u.x_v$,
$G=x_v.x_v$.
Now, of these three, $E$ and $G$ give the squared speeds of the $u$ and $v$ parameter curves of $x$. In some special cases (like surfaces of revolution), there exists a parametrization for which $E=1$. I am trying to understand what conditions can be imposed on $x$ such that both $E$ and $G$ equal 1.
A hand-wavy idea I had was to take the $u$ and $v$ parameter curves on $M$, and constructing $D$ by "unfurling" all the parameter curves to a plane. In that case, $u$ and $v$ become the arc-length parameters of the individual parameter curves, and consequently $E=G=1$. This may be possible at least for a subset of $M$, but I am not really sure if my intuition is right. Any help would be appreciated.

Answer 1

$E=G=1$ and $F=0$ if and only if the surface is locally isometric to the plane, and this holds if and only if the surface is flat (Gaussian curvature $K=0$).

(By the way, geodesic normal coordinates always give you coordinates with $E=1$ and $F=0$.)

It's easy to see why the condition is necessary (e.g., the formula for $K$ in terms of $E$ and $G$), but sufficiency will involve some differential equations; my favorite proof uses differential forms.