This survey starts with the historical landmarks leading to the study of principal
configurations on surfaces, their structural stability and further generalizations. Here it is …

We give a normal form of the cuspidal edge that uses only diffeomorphisms on the source
and isometries on the target. Using this normal form, we study differential geometric …

We define cuspidal curvature κ _c κ c (resp. normalized cuspidal curvature μ _c μ c) along
cuspidal edges (resp. at a swallowtail singularity) in Riemannian 3-manifolds, and show that …

We investigate the differential geometric ingredients for Whitney umbrella, which is known
as the only stable singularity of surface to 3-dimensional Euclidean space. We obtain …

We study the geometry of surfaces in R3 with corank 1 singularities. At a singular point we
define the curvature parabola using the first and second fundamental forms of the surface …

It is classically known that generic smooth maps of R^ 2 R 2 into R^ 3 R 3 admit only isolated
cross cap singularities. This suggests that the class of cross caps might be an important …

Along cuspidal edge singularities on a given surface in Euclidean 3-space R3, which can be
parametrized by a regular space curve ˆγ (t), a unit normal vector field ν is well-defined as a …

We study the geometry of germs of singular surfaces in R 3 whose parametrisations have an
A-singularity of Ae-codimension less than or equal to 3, via their contact with planes. These …

We obtain the topological configurations of the lines of curvature, the asymptotic and
characteristic curves on a cross-cap, in the domain of a parametrisation of this surface as …

Normal form of the swallowtail and its applications Page 1 International Journal of
Mathematics Vol. 29, No. 7 (2018) 1850046 (17 pages) c© World Scientific Publishing …