Geometric invariants and focal surfaces of spacelike curves in de Sitter space from a caustic viewpoint
| Autor: | Jiajing Miao, Haiming Liu |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Physics
focal surface De Sitter space Singularity theory lcsh:Mathematics General Mathematics Astrophysics::Instrumentation and Methods for Astrophysics lcsh:QA1-939 Space (mathematics) caustic General Relativity and Quantum Cosmology de sitter space invariant De Sitter universe singularity theory Gravitational singularity Focal surface Caustic (optics) Invariant (mathematics) Mathematical physics |
| Zdroj: | AIMS Mathematics, Vol 6, Iss 4, Pp 3177-3204 (2021) |
| ISSN: | 2473-6988 |
| DOI: | 10.3934/math.2021192 |
| Popis: | The focal surface of a generic space curve in Euclidean $ 3 $-space is a classical subject which is a two dimensional caustic and has Lagrangian singularities. In this paper, we define the first de Sitter focal surface and the second de Sitter focal surface of de Sitter spacelike curve and consider their singular points as an application of the theory of caustics and Legendrian dualities. The main results state that de Sitter focal surfaces can be seen as two dimensional caustics which have Lagrangian singularities. To characterize these singularities, a useful new geometric invariant $ \rho(s) $ is discovered and two dual relationships between focal surfaces and spacelike curve are given. Three examples are used to demonstrate the main results. |
| Databáze: | OpenAIRE |
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