Normal form of D4-plus singularities of fronts and its applications

We construct a form of the $D_4^{+}$-singularity of fronts in $R^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we calculate differential geometric invariants near the $D_4^{+}$-singularity, and give a Gauss-Bonnet type theorem for fronts allowing to have this singularity.