I wonder whether it is always possible to bring a 2D Riemannian metric to a diagonal form with determinant one by changing the coordinates, i.e. for the line element
$$
 ds^2 = A(x,y)\, dx^2 + B(x,y)\, dy^2
$$
to obtain $A(x,y)\, B(x,y) = 1$ everywhere.


It is known that one can bring the metric to a diagonal and conformally flat form
$$
 ds^2 = C(x,y)\, (dx^2 + dy^2)
$$
but it is not enough yet.



2D-metric to diagonal form with determinant 1
Locally, this is always possible. Constructing such a coordinate system is equivalent to solving a first-order hyperbolic PDE system for two unknowns of two variables, so it always has local smooth solutions.


Here is a sketch of how this can be proved: You want to find a $g$-orthonormal coframing $g = {\omega\_1}^2+{\omega\_2}^2$ and a function $\lambda$ so that $\mathrm{e}^\lambda\omega\_1$ and $\mathrm{e}^{-\lambda}\omega\_2$ are both closed. Then they can be written as $\omega\_1 = e^{-\lambda}\,\mathrm{d}x$ and $\omega\_2 = e^{\lambda}\,\mathrm{d}y$ for some functions $x$ and $y$ and these are the coordinates you want.


Now, if you let $g = {\eta\_1}^2+{\eta\_2}^2$ be any local orthonormal coframing, then, up to a possible change of sign of $\omega\_2$ (which won't affect the argument), the general orthonormal coframing of $g$ can be written as
$$
\omega\_1 = \cos\theta\,\eta\_1 -\sin\theta\,\eta\_2
\quad\text{and}\quad 
\omega\_2 = \sin\theta\,\eta\_1 +\cos\theta\,\eta\_2\tag1
$$
for some arbitrary function $\theta$. Now consider the equations
$$
\mathrm{d}(e^\lambda\,\omega\_1) = \mathrm{d}(e^{-\lambda}\,\omega\_2) = 0.\tag2
$$
These constitute two first-order, quasi-linear equations for the two unknown function $\theta$ and $\lambda$. It is easy to show that they constitute a hyperbolic system. (The characteristics of a given solution are given by $\omega\_1\pm\omega\_2=0$.)


In the smooth category, the local existence of smooth solutions of such systems is well-known. The non-characteristic initial value problem (always locally solvable) is specified as follows: First choose an embedded smooth curve $C$ in the domain of the coframing $\eta$ and choose a smooth function $\theta$ along $C$ subject to the open condition that $\omega\_1+\omega\_2$ and $\omega\_1-\omega\_2$ do not vanish when pulled back to $C$. Then choose an arbitrary function $\lambda$ along $C$. Then there will be an open neighborhood $U$ of $C$ to which $\theta$ and $\lambda$ extend uniquely to $U$ satisfying $(2)$.


