The conformal rotation number (Q2345639)

As it is known, the rotation number of a planar closed curve is the total curvature divided by \(2\pi\). This is a regular homotopy invariant of the curve. The author generalizes the rotation number to a curve on a closed surface using conformal geometry of the ambient surface. This conformal rotation number is not integral in general. The author shows that the fractional part is relevant to harmonic 1-forms of the surface.