Slant Curves in Contact Lorentzian Manifolds with CR Structures
In this paper, we first find the properties of the generalized Tanaka−Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the <inline-formula> <math display="inline"> <semantics> <mover> <mo&g...
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| Format: | Article |
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MDPI AG
2020-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/1/46 |
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| author | Ji-Eun Lee |
| author_facet | Ji-Eun Lee |
| author_sort | Ji-Eun Lee |
| collection | DOAJ |
| description | In this paper, we first find the properties of the generalized Tanaka−Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the <inline-formula> <math display="inline"> <semantics> <mover> <mo>∇</mo> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mo>≤</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, there does not exist a non-geodesic slant Frenet curve satisfying the <inline-formula> <math display="inline"> <semantics> <mover> <mo>∇</mo> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>-Jacobi equations for the <inline-formula> <math display="inline"> <semantics> <mover> <mo>∇</mo> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>-geodesic vector fields in <i>M</i>. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mn>1</mn> <mn>3</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mn>2</mn> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. |
| first_indexed | 2024-12-16T07:10:24Z |
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| id | doaj.art-90bcac740210487789caf435f7826dc5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-16T07:10:24Z |
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| spelling | doaj.art-90bcac740210487789caf435f7826dc52022-12-21T22:39:56ZengMDPI AGMathematics2227-73902020-01-01814610.3390/math8010046math8010046Slant Curves in Contact Lorentzian Manifolds with CR StructuresJi-Eun Lee0Institute of Basic Science, Chonnam National University, Gwangju 61186, KoreaIn this paper, we first find the properties of the generalized Tanaka−Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the <inline-formula> <math display="inline"> <semantics> <mover> <mo>∇</mo> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mo>≤</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, there does not exist a non-geodesic slant Frenet curve satisfying the <inline-formula> <math display="inline"> <semantics> <mover> <mo>∇</mo> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>-Jacobi equations for the <inline-formula> <math display="inline"> <semantics> <mover> <mo>∇</mo> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>-geodesic vector fields in <i>M</i>. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mn>1</mn> <mn>3</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mn>2</mn> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/8/1/46slant curvesjacobi equationcr structurelorentzian sasakian space forms |
| spellingShingle | Ji-Eun Lee Slant Curves in Contact Lorentzian Manifolds with CR Structures Mathematics slant curves jacobi equation cr structure lorentzian sasakian space forms |
| title | Slant Curves in Contact Lorentzian Manifolds with CR Structures |
| title_full | Slant Curves in Contact Lorentzian Manifolds with CR Structures |
| title_fullStr | Slant Curves in Contact Lorentzian Manifolds with CR Structures |
| title_full_unstemmed | Slant Curves in Contact Lorentzian Manifolds with CR Structures |
| title_short | Slant Curves in Contact Lorentzian Manifolds with CR Structures |
| title_sort | slant curves in contact lorentzian manifolds with cr structures |
| topic | slant curves jacobi equation cr structure lorentzian sasakian space forms |
| url | https://www.mdpi.com/2227-7390/8/1/46 |
| work_keys_str_mv | AT jieunlee slantcurvesincontactlorentzianmanifoldswithcrstructures |