  The modulus metric (also called the capacity metric) on a domain $D\subset
\mathbb{R}^n$ can be defined as $\mu_D(x,y)=\inf\{{\mbox{cap}}\,(D,\gamma)\}$,
where ${\mbox{cap}}\,(D,\gamma)$ stands for the capacity of the condenser
$(D,\gamma)$ and the infimum is taken over all continua $\gamma\subset D$
containing the points $x$ and $y$. It was conjectured by J. Ferrand, G. Martin
and M. Vuorinen in 1991 that every isometry in the modulus metric is a
conformal mapping. In this note, we confirm this conjecture and prove new
geometric properties of surfaces that are spheres in the metric space
$(D,\mu_D)$.
