The modulus of a family of curves on an abstract surface over a spherical ring

We obtain a two-sided estimate for the modulus of the family of all locally rectifiable curves joining two concentric spheres on a so-called abstract surface. The last notion means that, for a given curve and a point on it, the length element of the curve at this point depends on the direction of movement along the curve; in addition, the volume element is generated by some weight function.