The duality between singular points and inflection points on wave fronts

Abstract

In the previous paper, the authors gave criteria for $A_{k+1}$-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of $2$-inflection points (i.e. godron points) on an immersed surface in the real projective space is equal to the Euler number of $M_{-}$. Here $M^{2}$ is a compact orientable 2-manifold, and $M_{-}$ is the open subset of $M^{2}$ where the Hessian of $f$ takes negative values. This is a generalization of Bleecker and Wilson's formula [3] for immersed surfaces in the affine $3$-space.