Gauss-Bonnet type formulas for frontal bundles over surfaces with boundary and their applications

We define a frontal bundle by imposing a compatibility condition on two types of coherent tangent bundles over a surface with boundary. Since it is known that there are two Gauss-Bonnet type formulas for coherent tangent bundles, we obtain four Gauss-Bonnet type formulas for frontal bundles. Furthermore, since two coherent tangent bundles over the surface are related to each other by the compatibility condition, an appropriate combination of these Gauss-Bonnet type formulas leads to new formulas that relate a frontal to its unit normal vector field using the Gaussian curvature, the Euler characteristic, etc. If the extrinsic curvature of a frontal is non-zero bounded, these formulas lead to formulas that relate the number of singular points to the Euler characteristic.