Some properties of the modulus of a family of curves on an abstract surface

Consider a domain in Euclidean space whose volume element is induced by some weight function, while the arclength element of a curve at a point depends not only on the point, but also on the direction of motion along the curve. In this case we say that an abstract surface is defined over this domain. We prove a version of symmetry principle for the modulus of a family of curves on an abstract surface. In the weighted case we establish that the modulus is continuous when the arclength element is given in the isothermal coordinates.