Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula>
A translation curve in a Thurston space is a curve such that for given unit vector at the origin, the translation of this vector is tangent to the curve in every point of the curve. In most Thurston spaces, translation curves coincide with geodesic lines. However, this does not hold for Thurston spa...
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| Format: | Article |
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MDPI AG
2023-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/6/1533 |
| _version_ | 1797610275018375168 |
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| author | Zlatko Erjavec |
| author_facet | Zlatko Erjavec |
| author_sort | Zlatko Erjavec |
| collection | DOAJ |
| description | A translation curve in a Thurston space is a curve such that for given unit vector at the origin, the translation of this vector is tangent to the curve in every point of the curve. In most Thurston spaces, translation curves coincide with geodesic lines. However, this does not hold for Thurston spaces equipped with twisted product. In these spaces, translation curves seem more intuitive and simpler than geodesics. In this paper, geodesics and translation curves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> space are classified and the curvature properties of translation curves are investigated. |
| first_indexed | 2024-03-11T06:13:05Z |
| format | Article |
| id | doaj.art-2321d2665f174ec4a84ba6b1980b87aa |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T06:13:05Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2321d2665f174ec4a84ba6b1980b87aa2023-11-17T12:29:58ZengMDPI AGMathematics2227-73902023-03-01116153310.3390/math11061533Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula>Zlatko Erjavec0Faculty of Organization and Informatics, University of Zagreb, HR-42000 Varaždin, CroatiaA translation curve in a Thurston space is a curve such that for given unit vector at the origin, the translation of this vector is tangent to the curve in every point of the curve. In most Thurston spaces, translation curves coincide with geodesic lines. However, this does not hold for Thurston spaces equipped with twisted product. In these spaces, translation curves seem more intuitive and simpler than geodesics. In this paper, geodesics and translation curves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> space are classified and the curvature properties of translation curves are investigated.https://www.mdpi.com/2227-7390/11/6/1533geodesictranslation curvesolvable Lie group<named-content content-type="equation"><inline-formula> <mml:math id="mm310000"> <mml:semantics> <mml:msubsup> <mml:mi>Sol</mml:mi> <mml:mn>0</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content> space |
| spellingShingle | Zlatko Erjavec Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> Mathematics geodesic translation curve solvable Lie group <named-content content-type="equation"><inline-formula> <mml:math id="mm310000"> <mml:semantics> <mml:msubsup> <mml:mi>Sol</mml:mi> <mml:mn>0</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content> space |
| title | Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> |
| title_full | Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> |
| title_fullStr | Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> |
| title_full_unstemmed | Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> |
| title_short | Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula> |
| title_sort | geodesics and translation curves in inline formula math display inline semantics msubsup mi sol mi mn 0 mn mn 4 mn msubsup semantics math inline formula |
| topic | geodesic translation curve solvable Lie group <named-content content-type="equation"><inline-formula> <mml:math id="mm310000"> <mml:semantics> <mml:msubsup> <mml:mi>Sol</mml:mi> <mml:mn>0</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content> space |
| url | https://www.mdpi.com/2227-7390/11/6/1533 |
| work_keys_str_mv | AT zlatkoerjavec geodesicsandtranslationcurvesininlineformulamathdisplayinlinesemanticsmsubsupmisolmimn0mnmn4mnmsubsupsemanticsmathinlineformula |