Take a tangent vector field with isolated singularities on a compact smooth Riemannian surface ( 2 dimensional manifold without boundary). Divide v by its norm to get a field of unit vectors with isolated discontinuities. Around each singularity chose a small open disc. The tangent circle bundle over this disc is a solid torus. Removing all of the solid tori leaves the manifold,(M - the discs)x S1, and the unit circle bundle can be reconstituted by gluing the solid tori back in along their boundaries. I am trying to understand what the gluing map is. I think it is just gluing the bounding tori together by the map defined by the vector field. This map will wind around one circle of the torus once while winding around the second circle n-times where n is the index of the vector field. Is this right? What is the proof if it is right?