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We define a frontal bundle by imposing a compatibility condition on two typesof coherent tangent bundles over a surface with boundary. Since it is knownthat there are two Gauss-Bonnet type formulas for coherent tangent bundles, weobtain four Gauss-Bonnet type formulas for frontal bundles. Furthermore, sincetwo coherent tangent bundles over the surface are related to each other by thecompatibility condition, an appropriate combination of these Gauss-Bonnet typeformulas leads to new formulas that relate a frontal to its unit normal vectorfield using the Gaussian curvature, the Euler characteristic, etc. If theextrinsic curvature of a frontal is non-zero bounded, these formulas lead toformulas that relate the number of singular points to the Euler characteristic.

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