Singularities of Indefinite Hypersurfaces and Indefinite Surfaces along Timelike Curves in Nullcone with Index Two

Abstract

In this paper, a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves located in nullcone in 4-dimensional semi-Euclidean space with index 2 are discussed. Using the unfolding theory in singularity theory, the singularities of the indefinite hypersurfaces and the indefinite surfaces are classified and the different kinds of singularities are estimated by means of a geometric invariant σ. Meanwhile, the definition of osculating nullcone is presented; the study shows that the differential geometric invariant σ of timelike curves measured also the order of the contact between a timelike curve and a osculating nullcone ONCv0⁎. Finally, some relevant counterexamples are indicated.