Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space

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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Autor principal: Fatma Bulut
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
<i>k</i>-type helices
(<i>k</i>,<i>m</i>)-type slant helices
extended Darboux frame field
Minkowski space
anti-symmetric matrix
Mathematics
QA1-939
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spelling oai:doaj.org-article:d92bb62fed964296bd5c394774c9324b2021-11-25T19:07:25ZSlant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space10.3390/sym131121852073-8994https://doaj.org/article/d92bb62fed964296bd5c394774c9324b2021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2185https://doaj.org/toc/2073-8994In 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.Fatma BulutMDPI AGarticle<i>k</i>-type helices(<i>k</i>,<i>m</i>)-type slant helicesextended Darboux frame fieldMinkowski spaceanti-symmetric matrixMathematicsQA1-939ENSymmetry, Vol 13, Iss 2185, p 2185 (2021)
institution DOAJ
collection DOAJ
language EN
topic <i>k</i>-type helices
(<i>k</i>,<i>m</i>)-type slant helices
extended Darboux frame field
Minkowski space
anti-symmetric matrix
Mathematics
QA1-939
spellingShingle <i>k</i>-type helices
(<i>k</i>,<i>m</i>)-type slant helices
extended Darboux frame field
Minkowski space
anti-symmetric matrix
Mathematics
QA1-939
Fatma Bulut
Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space
description 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.
format article
author Fatma Bulut
author_facet Fatma Bulut
author_sort Fatma Bulut
title Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space
title_short Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space
title_full Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space
title_fullStr Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space
title_full_unstemmed Slant Helices of (<i>k</i>,<i>m</i>)-Type According to the ED-Frame in Minkowski 4-Space
title_sort slant helices of (<i>k</i>,<i>m</i>)-type according to the ed-frame in minkowski 4-space
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/d92bb62fed964296bd5c394774c9324b
work_keys_str_mv AT fatmabulut slanthelicesofikiimitypeaccordingtotheedframeinminkowski4space
_version_ 1718410309158305792

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