Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space
In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this...
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| Formato: | article |
| Lenguaje: | EN |
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MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/d92bb62fed964296bd5c394774c9324b |
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| Sumario: | In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ε</mi><mn>1</mn></msub><mo>=</mo><msub><mi>ε</mi><mn>2</mn></msub><mo>=</mo><msub><mi>ε</mi><mn>3</mn></msub><mo>=</mo><msub><mi>ε</mi><mn>4</mn></msub><mo>∈</mo><mrow><mo>{</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ε</mi><mn>1</mn></msub><mo>=</mo><msub><mi>ε</mi><mn>2</mn></msub><mo>=</mo><msub><mi>ε</mi><mn>3</mn></msub><mo>=</mo><msub><mi>ε</mi><mn>4</mn></msub><mo>∈</mo><mrow><mo>{</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula> in four-dimensional Minkowski space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="double-struck">E</mi><mn>1</mn><mn>4</mn></msubsup></mrow></semantics></math></inline-formula>. In addition, we present some characterizations of slant helices and determine (<i>k</i>,<i>m</i>)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space. |
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