Let $g$ be a metric of arbitrary signature on $\Bbb R^n$. If the Riemann tensor of $g$ vanishes identically, is there a diffeomorphism $f:\Bbb R^n\to\Bbb R^n$ with $g=f^*e$, where $e$ is the standard metric with the same signature as $g$?
In the Riemannian case, assuming $(\Bbb R^n,g)$ is complete, this is well known from the classification of space forms. I was unable to show that $(\Bbb R^n,g)$ is always complete in the Riemannian case, and I don't know if it's true or not. I also know that this is always true in a local sense, but I am looking for a global isometry.