Singularities Of Hypersurfaces Along Framed Null Cartan Curves In Minkowski 4-Space

In this work,we investigate the singularities of two lightlike hypersurfaces,a lightlike surface,a spacelike surface and two curves associated with the singular null Cartan curve in Minkowski 4-space.In the course of the study,we establish a Cartan Frenet type equation and three important invariants associated with the null Cartan curve.According to unfolding theory,we characterize the singularities of the lightlike hypersurfaces,the surfaces and the curves by means of geometric invariants and then singularities types can be determined by the three geometric invariants.Using the theory of Legendrian dualities,it is found that the singularities of the surface is Δ2-dual to the tangent trajectory L(t)of the null Cartan curve in H03,and the surface is locally diffeomorphic to the swallowtail SW or cuspidal edge CE under different conditions.It is also shown that there exist deep relationships between the singularities of the surface and the curve and the order of contact between L(t)and elliptic quadric εQ(v0,-1)or the order of contact between L(t)and spacelike hyperplane HP(v0,-1)in H03.Finally,we present several examples to describe the main results.