A generalization of Zakalyukin's lemma, and symmetries of surface singularities

Atsufumi Honda, Kosuke Naokawa, Kentaro Saji, Masaaki Umehara, and Kotaro Yamada

Journal of Singularities
volume 25 (2022), 299-324

Received: 19 January 2021. Received in revised form: 10 July 2021.

DOI: 10.5427/jsing.2022.25m

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Abstract:

Zakalyukin's lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.

2020 Mathematical Subject Classification:

53A05, 57R45, 57R3

Key words and phrases:

singularity, wave front, cuspidal edge, first fundamental form

Author(s) information:

Atsufumi Honda
Department of Applied Mathematics
Faculty of Engineering
Yokohama National University
79-5 Tokiwadai, Hodogaya Yokohama 240-8501, Japan
honda-atsufumi-kp@ynu.ac.jp

Kosuke Naokawa
Department of Computer Science
Faculty of Applied Information Science
Hiroshima Institute of Technology
2-1-1 Miyake, Saeki, Hiroshima, 731-5193, Japan
k.naokawa.ec@cc.it-hiroshima.ac.jp

Kentaro Saji
Department of Mathematics
Faculty of Science
Kobe University
Rokko, Kobe 657-8501
saji@math.kobe-u.ac.jp

Masaaki Umehara
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
Tokyo 152-8552, Japan
umehara@is.titech.ac.jp

Kotaro Yamada
Department of Mathematics
Tokyo Institute of Technology
Tokyo 152-8551, Japan
kotaro@math.titech.ac.jp