The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications
arXiv:1802.05557 · doi:10.1007/s12220-019-00197-0
Abstract
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.
arXiv comment28 pages, 8 figures
References in corpus (9)
- Singularities of flat fronts in hyperbolic 3-space
- The Wigner caustic on shell and singularities of odd functions
- The improved isoperimetric inequality and the Wigner caustic of planar ovals
- Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds
- Even Dimensional Improper Affine Spheres
- Singular points of the Wigner caustic and affine equidistants of planar curves
- The Constant Width Measure Set, the Spherical Measure Set and isoperimetric equalities for planar ovals
- Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space
- The Geometry of the Wigner Caustic and a Decomposition of a Curve Into Parallel Arcs