Resumo: In [2,3,4] 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 in [1]. We apply this theorem to investigate global properties of maps between surfaces with boundary. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes) and extended evolutoids front for planar closed hedghehogs [1,5]. In particular, we find integral formulas involving complicated relations of a width and a radius of curvature of a planar hedgehog.

[1] W. Domitrz, M. Zwierzyński: ”The Gauss-Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications”, J Geom Anal 30, 3243–3274 (2020).
[2] K. Saji, M. Umeraha, K. Yamada: Behavior of corank-one singular points on wave fronts, Kyushu Journal of Mathematics, Vol. 62 (2008), 259–280.
[3] K. Saji, M. Umeraha, K. Yamada, The geometry of fronts, Annals of Mathematics, 169 (2009), 491–529.
[4] K. Saji, M. Umehara, K. Yamada, Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts, J. Geom. Anal. 22 (2012), no. 2, 383–409. [5] M. Zwierzyński: ”The Extended Front of Evolutoids and the Singular Evolutoids Set”, arXiv