This is a joint work with Michal Zwierzynski.
K. Saji, M. Umehara, K. Yamada (Annals of Mathematics, 169, 2009) proved the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary). 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 rosette.