Singularities of null surfaces of null Cartan curves in three-dimensional anti-de Sitter space
The authors study the geometric properties of a kind of null surfaces associated to null Cartan curves in 3-dimensional anti-de Sitter space from the singularity theory approach in the present paper. The authors mention and believe that the geometric informations in anti-de Sitter space are more rich than in hyperbolic space. Initially, they describe the local differential geometry of null Cartan curves in anti-de Sitter space. The authors define the binormal Gauss indicatrix \(\mathbf{B}\) and principal Gauss indicatrix \(\phi\) of the null Cartan curve \(\gamma\). By using Gauss indicatrixes, they define a null surface \(\Phi\) associated to the null Cartan curve \(\gamma\) and call it the anti-de Sitter null Cartan surface. The authors also introduce two new invariants of the null Cartan curve \(\gamma\) in a three dimensional anti-de Sitter space \(H_{1}^{3}=\{\text{x}\in \mathbb{R}_{2}^{4} | <\text{x}, \text{x}>=-1\}\) by \(\delta (s)=\frac{\kappa^{\prime}}{(\kappa^{2}-1)^{3/2}}\) and \(\eta (s) = \delta^{\prime}(s)-\frac{1}{\sqrt{\kappa^{2}-1}}\) which are useful to describe the geometric meanings of singularities of the anti-de Sitter null Cartan surfaces. To relate the binormal Gauss indicatrix \(\mathbf{B}\) with the models, they define a family of functions named binormal height function \(H(s,\text{v})=<\mathbf{B}(s),\text{v}>\), where \(H:I\times H_{1}^{3}\rightarrow \mathbb{R}\). They also give the explanation of geometric meanings of the invariants \(\kappa (s)\), \(\delta (s)\) and \(\eta (s)\). The authors provide the classification of singularities of anti-de Sitter null Cartan surfaces as applications of singularity theory of functions by taking advantage of the unfolding theory of functions of one variable. Finally, the authors show that the anti-de Sitter null Cartan surface is locally diffeomorphic to the cuspidal edge (under 3-point contact) or the swallowtail (under 4-point contact).
scientific article; zbMATH DE number 6822554
scientific article