Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime
This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&q...
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2025-04-01
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| author | Xiaoming Fan Yongsheng Zhu Haijing Pan |
| author_facet | Xiaoming Fan Yongsheng Zhu Haijing Pan |
| author_sort | Xiaoming Fan |
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| description | This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msub><mi>H</mi><mi mathvariant="bold-italic">β</mi></msub></mrow></semantics></math></inline-formula>, a critical lightlike surface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msub><mi>S</mi><mi mathvariant="bold-italic">β</mi></msub></mrow></semantics></math></inline-formula>, and a degenerate curve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msub><mi>C</mi><mi mathvariant="bold-italic">β</mi></msub></mrow></semantics></math></inline-formula>, with dimensions decreasing from 3D to 1D. Using singularity theory, we identify a novel geometric invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> that governs the emergence of specific singularity types, including <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>×</mo><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>W</mi><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>F</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>B</mi><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>-cusp. These singularities exhibit increasing degeneracy as the hierarchy progresses, with contact orders between the lightlike hyperplane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><msubsup><mi>S</mi><msub><mi>t</mi><mn>0</mn></msub><mi>L</mi></msubsup></mrow></semantics></math></inline-formula> and the curve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">β</mi></semantics></math></inline-formula> systematically intensifying. An explicit example demonstrates the construction of these objects and validates the theoretical results. This work establishes a systematic connection between null Cartan curves, stratified singularities, and contact geometry. |
| format | Article |
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| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-04-01 |
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| spelling | doaj-art-965f54b0aa844f3983be51dddf078ed72025-08-20T02:17:25ZengMDPI AGAxioms2075-16802025-04-0114427910.3390/axioms14040279Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski SpacetimeXiaoming Fan0Yongsheng Zhu1Haijing Pan2School of Teacher Education, Harbin Normal University, Harbin 150025, ChinaSchool of Mathematical Sciences, Harbin Normal University, Harbin 150025, ChinaSchool of Mathematical Sciences, Harbin Normal University, Harbin 150025, ChinaThis study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msub><mi>H</mi><mi mathvariant="bold-italic">β</mi></msub></mrow></semantics></math></inline-formula>, a critical lightlike surface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msub><mi>S</mi><mi mathvariant="bold-italic">β</mi></msub></mrow></semantics></math></inline-formula>, and a degenerate curve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msub><mi>C</mi><mi mathvariant="bold-italic">β</mi></msub></mrow></semantics></math></inline-formula>, with dimensions decreasing from 3D to 1D. Using singularity theory, we identify a novel geometric invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> that governs the emergence of specific singularity types, including <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>×</mo><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>W</mi><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>F</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>B</mi><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>-cusp. These singularities exhibit increasing degeneracy as the hierarchy progresses, with contact orders between the lightlike hyperplane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><msubsup><mi>S</mi><msub><mi>t</mi><mn>0</mn></msub><mi>L</mi></msubsup></mrow></semantics></math></inline-formula> and the curve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">β</mi></semantics></math></inline-formula> systematically intensifying. An explicit example demonstrates the construction of these objects and validates the theoretical results. This work establishes a systematic connection between null Cartan curves, stratified singularities, and contact geometry.https://www.mdpi.com/2075-1680/14/4/279lightlike hypersurfacesnull Cartan curvesMinkowski spacetimesingularitySwallowtail |
| spellingShingle | Xiaoming Fan Yongsheng Zhu Haijing Pan Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime Axioms lightlike hypersurfaces null Cartan curves Minkowski spacetime singularity Swallowtail |
| title | Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime |
| title_full | Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime |
| title_fullStr | Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime |
| title_full_unstemmed | Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime |
| title_short | Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime |
| title_sort | singularity analysis of lightlike hypersurfaces generated by null cartan curves in minkowski spacetime |
| topic | lightlike hypersurfaces null Cartan curves Minkowski spacetime singularity Swallowtail |
| url | https://www.mdpi.com/2075-1680/14/4/279 |
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