  In this article, we study the role of invariant distances in the Pick
interpolation problem. Given a Carath\'eodory hyperbolic domain $\Omega$ in
some $\mathbb{C}^m$, we have introduced a notion of an invariant object that
gives a necessary and sufficient condition for any Pick interpolation problem
to be solvable on $\Omega$. This invariant object plays the same role as that
of Carath\'eodory pseudodistance in the two-point Pick interpolation problem.
Furthermore, a full description of the invariant object is given when $\Omega$
is the open unit disc.
