The symmetry principle and nondegenerate families of curves on abstract surfaces

Consider a Euclidean space domain whose volume element is induced by some weight function, while the arclength element of a curve at an arbitrary 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. The article answers the question of whether a family of locally rectifiable curves of infinite length on an abstract surface has modulus zero. We establish sufficient conditions for the symmetry principle to hold on an abstract surface. Also we state the requirements allowing us to approximate the modulus of a family of curves on an abstract surface while restricting the class of admissible functions to bounded ones.