Geometry of $D_4^{-}$-front singularities and a Gauss-Bonnet type formula

We construct a form of the $D_4^{-}$-singularity of fronts in ${\mathbb{R}}^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we compute differential geometric invariants near the $D_4^{-}$-singularity, and give a Gauss-Bonnet type theorem for one-parameter generic fronts.