SINGULARITIES OF FLAT FRONTS IN HYPERBOLIC SPACE (vol 221, pg 303, 2005)

摘要

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.