We introduce null surfaces(or nullcone fronts)of pseudo-spherical spacelike framed curves in the three-dimensional anti-de Sitter space.These surfaces are formed by the light rays emitted from points on anti-de Sitter...We introduce null surfaces(or nullcone fronts)of pseudo-spherical spacelike framed curves in the three-dimensional anti-de Sitter space.These surfaces are formed by the light rays emitted from points on anti-de Sitter spacelike framed curves.We then classify singularities of the nullcone front of a pseudo-spherical spacelike framed curve and show how these singularities are related to the singularities of the associated framed curve.We also define a family of functions called the Anti-de Sitter distance-squared functions to explain the nullcone front of a pseudo-spherical spacelike framed curve as a wavefront from the viewpoint of the Legendrian singularity theory.We finally provide some examples to illustrate the results of this paper.展开更多
Anti de Sitter space is a maximally symmetric, vacuum solution of Einstein's field equation with an attractive cosmological constant, and is the hyperquadric of semi-Euclidean space with index 2. So it is meaningf...Anti de Sitter space is a maximally symmetric, vacuum solution of Einstein's field equation with an attractive cosmological constant, and is the hyperquadric of semi-Euclidean space with index 2. So it is meaningful to study the submanifold in semi-Euclidean 4-space with index 2. However, the research on the submanifold in semi-Euclidean 4-space with index 2 has not been found from theory of singularity until now. In this paper, as a generalization of the study on lightlike hypersurface in Minkowski space and a preparation for the further study on anti de Sitter space, the singularities of lightlike hypersurface and Lorentzian surface in semi- Euclidean 4-space with index 2 will be studied. To do this, we reveal the relationships between the singularity of distance-squared function and that of lightlike hypersurface. In addition some geometric properties of lightlike hypersurface and Lorentzian surface are studied from geometrical point of view.展开更多
文摘We introduce null surfaces(or nullcone fronts)of pseudo-spherical spacelike framed curves in the three-dimensional anti-de Sitter space.These surfaces are formed by the light rays emitted from points on anti-de Sitter spacelike framed curves.We then classify singularities of the nullcone front of a pseudo-spherical spacelike framed curve and show how these singularities are related to the singularities of the associated framed curve.We also define a family of functions called the Anti-de Sitter distance-squared functions to explain the nullcone front of a pseudo-spherical spacelike framed curve as a wavefront from the viewpoint of the Legendrian singularity theory.We finally provide some examples to illustrate the results of this paper.
基金supported by National Natural Science Foundation of China (Grant No.10871035)the New Century Excellent Talents in University of China (Grant No. 05-0319)
文摘Anti de Sitter space is a maximally symmetric, vacuum solution of Einstein's field equation with an attractive cosmological constant, and is the hyperquadric of semi-Euclidean space with index 2. So it is meaningful to study the submanifold in semi-Euclidean 4-space with index 2. However, the research on the submanifold in semi-Euclidean 4-space with index 2 has not been found from theory of singularity until now. In this paper, as a generalization of the study on lightlike hypersurface in Minkowski space and a preparation for the further study on anti de Sitter space, the singularities of lightlike hypersurface and Lorentzian surface in semi- Euclidean 4-space with index 2 will be studied. To do this, we reveal the relationships between the singularity of distance-squared function and that of lightlike hypersurface. In addition some geometric properties of lightlike hypersurface and Lorentzian surface are studied from geometrical point of view.
