Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula>
We investigate the class of helical hypersurfaces parametrized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">x</mi><mo>=</mo><mi mathvari...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-07-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/9/7/341 |
| _version_ | 1797587343420424192 |
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| author | Erhan Güler |
| author_facet | Erhan Güler |
| author_sort | Erhan Güler |
| collection | DOAJ |
| description | We investigate the class of helical hypersurfaces parametrized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">x</mi><mo>=</mo><mi mathvariant="fraktur">x</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula>, characterized by a light-like axis in Minkowski spacetime <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula>. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">x</mi></semantics></math></inline-formula>. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">x</mi></semantics></math></inline-formula>. We explore the conditions under which the curvatures of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">x</mi></semantics></math></inline-formula> possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T00:35:46Z |
| format | Article |
| id | doaj.art-ff62df8001114536af80dc8bf93f21b3 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-11T00:35:46Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-ff62df8001114536af80dc8bf93f21b32023-11-18T21:39:59ZengMDPI AGUniverse2218-19972023-07-019734110.3390/universe9070341Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula>Erhan Güler0Department of Mathematics, Faculty of Sciences, Bartın University, Kutlubey Campus, 74100 Bartın, TurkeyWe investigate the class of helical hypersurfaces parametrized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">x</mi><mo>=</mo><mi mathvariant="fraktur">x</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula>, characterized by a light-like axis in Minkowski spacetime <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula>. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">x</mi></semantics></math></inline-formula>. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">x</mi></semantics></math></inline-formula>. We explore the conditions under which the curvatures of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">x</mi></semantics></math></inline-formula> possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula>.https://www.mdpi.com/2218-1997/9/7/341Minkowski spacetimehelical hypersurfacelight-like axisGauss mapcurvature |
| spellingShingle | Erhan Güler Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula> Universe Minkowski spacetime helical hypersurface light-like axis Gauss map curvature |
| title | Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_full | Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_fullStr | Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_full_unstemmed | Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_short | Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">L</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_sort | differential geometry of the family of helical hypersurfaces with a light like axis in minkowski spacetime inline formula math display inline semantics msup mi mathvariant double struck l mi mn 4 mn msup semantics math inline formula |
| topic | Minkowski spacetime helical hypersurface light-like axis Gauss map curvature |
| url | https://www.mdpi.com/2218-1997/9/7/341 |
| work_keys_str_mv | AT erhanguler differentialgeometryofthefamilyofhelicalhypersurfaceswithalightlikeaxisinminkowskispacetimeinlineformulamathdisplayinlinesemanticsmsupmimathvariantdoublestrucklmimn4mnmsupsemanticsmathinlineformula |