In Saji et al. (J Math 62:259–280, 2008; Ann Math 169:491–529, 2009; J Geom Anal 222):383–409, 2012) 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 a special version of 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.

中文翻译：

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具有边界的表面上相干正切束的Gauss-Bonnet定理及其应用

在Saji等人中。（J Math 62：259–280，2008; Ann Math 169：491–529，2009; J Geom Anal 222）：383–409，2012）紧致定向表面（无边界）上相干切线束的Gauss-Bonnet公式）被证明。我们建立了带边界的紧致定向表面上相干切线束的高斯-邦尼定理。我们应用该定理来研究具有边界的曲面之间的地图的全局特性。作为结果的推论，我们获得了福田-石川定理的特殊版本。我们还研究了平面闭合非奇异刺猬（玫瑰花结）的仿射延伸波阵面的几何形状。特别是，我们发现边界上的总测地曲率和仿射扩展波前的总奇异曲率之间存在联系，这导致了玫瑰花结宽度函数的积分关系。