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We obtain a two-sided estimate for the modulus of the family of all locally recti able 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.

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