  In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over
compact oriented surfaces (without boundary) were proved. We establish the
Gauss-Bonnet theorem for coherent tangent bundles over compact oriented
surfaces with boundary. We apply this theorem to investigate global properties
of maps between surfaces with boundary. As a corollary of our results we obtain
Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave
fronts for planar closed non singular hedgehogs (rosettes). In particular, we
find a link between the total geodesic curvature on the boundary and the total
singular curvature of the affine extended wave front, which leads to a relation
of integrals of functions of the width of a resette.
