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Ak singlarities of wave fronts

Published online by Cambridge University Press:  01 May 2009

KENTARO SAJI
Affiliation:
Department of Mathematics, Faculty of Educaton Gifu University, Yanagido 1-1, Gifu 151-1193, Japan. e-mail: ksaji@gifu-u.ac.jp
MASAAKI UMEHARA
Affiliation:
Department of Mathematics, Graduate School of Science Osaka University, Toyonaka, Osaka 560-0043, Japan. e-mail: umehara@math.sci.osaka-u.ac.jp
KOTARO YAMADA
Affiliation:
Faculty of Mathematics, Kyushu UniversityFukuoka 812-8581, Japan. e-mail: kotaro@math.kyushu-u.ac.jp

Abstract

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 “kth 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 Rn+1 (resp. Cn+1) 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 Rn (resp. Cn) is right-left equivalent to the Ak-type singular point in Rn (resp. Cn). Moreover, we show that the restriction of a C-map f: RnRn 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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

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