Legendrian dual surfaces of a spacelike curve in the 3-dimensional lightcone. (English) Zbl 1503.53025
Summary: In this paper, we study Legendrian dual surfaces, lying in the 3-dimensional pseudo-(hyper)spheres of various radii in Lorentz-Minkowski 4-space, of a spacelike curve in the 3-dimensional lightcone. We point out that these surfaces are constructed by the extended Legendrian dualities between the 3-dimensional lightcone and these 3-dimensional pseudo-spheres. Moreover, we investigate the singularities of these surfaces and show the dualities of the singularities of a certain class of such a surface in the 3-dimensional lightcone. Furthermore, we deal with the singularities of all of these surfaces when they are non-flat.
MSC:
| 53A35 | Non-Euclidean differential geometry |
| 53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |
| 57R45 | Singularities of differentiable mappings in differential topology |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 58K99 | Theory of singularities and catastrophe theory |
Keywords:
singularities; Legendrian dual surfaces; space-like curve; 3-dimensional lightcone; Legendrian dualitiesReferences:
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