#### Vol. 294, No. 2, 2018

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Addendum: Singularities of flat fronts in hyperbolic space

### Masatoshi Kokubu, Wayne Rossman, Kentaro Saji, Masaaki Umehara and Kotaro Yamada

Vol. 294 (2018), No. 2, 505–509
##### Abstract

This is an addendum to the authors’ previous paper in which criteria for cuspidal edges and swallowtails on surfaces are given by applying the so-called Zakalyukin’s lemma. The original statement in Zakalyukin’s paper assumed the properness of the mappings. However, the lemma in the appendix of our paper did not assume properness. Recently, we noticed that the proof given in the appendix was implicitly relying on properness. In this addendum, we prove that mappings satisfying the criteria of cuspidal edges and swallowtails have properness. Consequently, the criteria are clarified.

##### Keywords
wave front, cuspidal edge, swallowtail
##### Mathematical Subject Classification 2010
Primary: 53C42, 57R45
Secondary: 53A05, 53A99