Lightcone dual surfaces and hyperbolic dual surfaces of spacelike curves in de Sitter 3-space (Q2812325)

If the pseudo scalar product in \(\mathbb R^4\) is defined as \(\langle x,y\rangle=-x_1y_1+x_2y_2+x_3y_3+x_4y_4\), then the pair \(\mathbb R^4_1=(\mathbb R^4,\langle,\rangle)\) is called the Minkowski spacetime. The de Sitter 3-space is \(S^3_1=\{x\in\mathbb R^4_1;\;\langle x,x\rangle=1\}\). A curve \(\gamma:I\to S^3_1\) is called a space-like curve if its velocity satisfies \(\langle\dot\gamma(t),\dot\gamma(t)\rangle>0\). The future lightcone at the origin is \(LC^3_+=\{x;\;\langle x,x\rangle, x_1>0\}\). In the theory of relativity, the future lightcone of an event is the boundary of its causal future in Minkowski spacetime. Any simply connected 2-dimensional Riemannian manifold can be isometrically immersed in a lightcone in Minkowski 4-space and a famous global geometry property is that a compact space-like surface in a lightcone is diffeomorphic to the 2-dimensional sphere \(S^2\). In [J. Math. Anal. Appl. 325, No. 2, 1171--1181 (2007; Zbl 1108.53011)], the first author studied surfaces in the light-like cone and showed the importance of the relations between the conformal transformation group and the Lorentzian group of \(\mathbb R^n_1\), the submanifolds of the Riemannian sphere \(S^n\) and the submanifolds of the lightcone \(LC^{n+1}\).NEWLINENEWLINEIn this paper, the authors investigate generic singularities of lightcone dual surfaces, which are space-like surfaces in the lightcone, and hyperbolic dual surfaces generated by space-like curves in de Sitter space. There are two kinds of space-like dual surfaces of spacelike curves. One is the dual of the space-like curve of the de Sitter 3-space embedded in the lightcone, and another is the dual of the space-like curve of the de Sitter 3-space embedded in the hyperbolic space. The authors give a classification of the singularities of lightcone dual surfaces and hyperbolic dual surfaces for generic space-like curves in de Sitter 3-space and illustrate the results with examples.