Gaussian surfaces and nullcone dual surfaces of null curves in a three-dimensional nullcone with index 2

In this paper, we consider the null curves in the 3-nullcone with index 2 and we investigate these curves in the framework of the theory of Legendrian dualities between nullcones. The sufficient and necessary conditions for the classifications of the singularities of both Gaussian surfaces and those nullcone dual surfaces that are associated with a null curve are given; these conditions are closely related to several new geometric invariants. In addition, we reveal the relationships between these geometric invariants and the order of contact for the 2nd principal normal curve n(s) of a null curve γ(s) with quadratic surfaces. Finally, two examples, namely, a Gaussian surface and a nullcone dual surface, are used to demonstrate our theoretical results.