Consider a Riemannian manifold $M$ with a metric $g$. For two points $x,y \in M$ the geodesic distance $d(x,y)$ is defined in the usual way.
I would like to know if there is a formula expressing how $d(x,y)$ transforms under a Weyl transformation $$ g(z) \mapsto e^{2 \sigma(z)} g(z). $$ Is there such a formula for arbitrary dimension of $M$, or at least for dimension 2?