  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.
