# 2D-metric to diagonal form with determinant 1

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.

• If the conformal factor factorizes, $C(x,y) = C_1(x) C_2(y)$, then the solution is easy. But in general? May 28, 2021 at 19:31
• Crossposted from physics.stackexchange.com/q/639706/2451 May 29, 2021 at 11:19

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 $$(2)$$.

• Thank you @RobertBryant! I didn't look at that in the language of differential forms which seems to be superior here (over PDEs). I need to check how it works in practice on some examples ;) This question comes from studying the Dirac equation in curved 2D space where the condition $\det(g) = 1$ would make some important simplifications possible. Jun 1, 2021 at 13:05