Criteria for D_4 singularities of wave fronts

We give useful and simple criteria for determining D_4 singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near D_4^+ singularities.Comment: 11 pages with 4 figure

Singularities of Blaschke normal maps of convex surfaces

We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.Comment: 5 pages Version 3; to appear in Comptes Rendus Mathematiqu

Criteria for Morin singularities into higher dimensions (Theory of singularities of smooth mappings and around it)

"Theory of singularities of smooth mappings and around it". November 25～29, 2013. edited by Takashi Nishimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We give criteria for Morin singularities into higher dimensions. As an application, we study the number of mathcal{A}-isotopy classes of Morin singularities