A K Singlarities of Wave Fronts

In this paper, we discuss the recognition problem for Ak-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors’ previous work “the geometry of fronts” for surfaces. The crucial point to prove our criteria for Ak-singularities is to introduce a suitable parametrization of the singularities called the “k-th KRSUY-coordinates” (see Section 3). Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a real (resp. complex) hypersurface (as a wave front) in R (resp. C) is differentiably (resp. holomorphically) right-left equivalent to the Ak+1-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane R (resp. C) is right-left equivalent to the Ak-type singular point in R (resp. C). Moreover, we show that the restriction of a C-map f : R → R to its Morin singular set gives a wave front consisting of only Ak-type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.