I am trying to show that there does not exist a Riemann metric $g$ on $\mathbb{R}^2$ which induces the distance $d_\infty(p,q)=\|p-q\|_\infty$, but am not entirely sure how to procede.
My first attempt was to use that $\mathbb{R}^2$ with $d_\infty$ is a complete metric space and thus if there exists such a metric a complete riemannian manifold so there exists a minimal geodesic between any two points $p,q$, and use this to somehow arrive at a contradiction. The somehow part is unfortunately evading me...
Any ideas how to procede, or alternative approaches?
The boundary of the unit ball in this metric is not a submanifold, because actually it is a square. Riemann metric is smooth, so unit ball should be a submanifold.