Whitney-type Formula for Non-null-homotopic Curves on Aspherical Surfaces

In an earlier paper, I defined a new winding number of regular closed curves on complete euclidean/hyperbolic surfaces and showed that this winding number, together with the free homotopy class, determines the regular homotopy class. In this paper, I give a Whitney-type formula for the winding number of non-null-homotopic generic regular closed curves on surfaces with a complete euclidean or hyperbolic structure, generalizing the formula for curves on a torus by Tanio and Kobayashi.