A k singlarities of wave fronts

Abstract

A k SINGLARITIES OF WAVE FRONTS.KENTARO SAJI, MASAAKI UMEHARA, AND KOTARO YAMADA Dedicated to Professor Yoshiaki Maeda on the occasion of his sixtieth birthday.Abstract. In this paper, we discuss the recognition problem for A k -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 A k -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 n+1 (resp. C n+1 ) is differentiably (resp. holomorphically) right-left equivalent to the A k+1 -type singular point if and only if the linear projection of the singular set around p into a generic hyperplane R n (resp. C n ) is right-left equivalent to the A k -type singular point in R n (resp. C n ). Moreover, we show that the restriction of a C ∞ -map f : R n → R n to its Morin singular set gives a wave front consisting of only A k -type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

“…Simple criteria for cuspidal edge and swallowtail were given by Kokubu, Rossman, Saji, Umehara and Yamada [28]. Other criteria for singularities of frontals are investigated in [13,32,22]. Here, we briefly review the criteria of frontals.…”

Whitney umbrellas and swallowtails

“…Hence, is a swallowtail by [Saji et al 2009a, Corollary 2.5], and we have proved that Ᏽ : ᐃ → is well-defined.…”

“…Then, G has the form of (3) from Section 1, which is a normalized swallowtail. Since G is normalized, ∂/∂ x is the null vector field for G defined in [Kokubu et al 2005;Saji et al 2009a], that is,…”

“…Since is normalized, ∂/∂ x is the null vector field. Since is a swallowtail, Lemma 3.2 follows from [Saji et al 2009a, Corollary 2.5].…”

Orientability of linear Weingarten surfaces, spacelike CMC‐1 surfaces and maximal surfaces

“…For a given co‐orientable front f : M 2 → H 3 , the zig‐zag representation is induced (cf. 24–26), which is invariant under the deformation of f as a wave front, and holds for the unit normal field ν: M 2 → S 3 1 .…”