Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity
This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a <i>n</i>-sided hyperbolic polygon <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" d...
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MDPI AG
2022-09-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/10/1979 |
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| author | Emmanuele Battista Giampiero Esposito |
| author_facet | Emmanuele Battista Giampiero Esposito |
| author_sort | Emmanuele Battista |
| collection | DOAJ |
| description | This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a <i>n</i>-sided hyperbolic polygon <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">P</mi></semantics></math></inline-formula>, which is the Euclidean closure of the hyperbolic plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">H</mi></semantics></math></inline-formula>, bounded by <i>n</i> hyperbolic geodesic segments. The polygon <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">P</mi></semantics></math></inline-formula> is built by considering the unique geodesic that connects the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula> vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>z</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo>,</mo><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>z</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>z</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. The geodesics that link the vertices are Euclidean semicircles centred on the real axis. The vector normal to the geodesic linking two consecutive vertices is evaluated and turns out to be discontinuous. Within the framework of elliptic geometry, we solve the geodesic equation and construct a geodesic triangle. Additionally in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined. |
| first_indexed | 2024-03-09T19:26:52Z |
| format | Article |
| id | doaj.art-79228f52b4324045aefc5ae77e80a90d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T19:26:52Z |
| publishDate | 2022-09-01 |
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| series | Symmetry |
| spelling | doaj.art-79228f52b4324045aefc5ae77e80a90d2023-11-24T02:50:01ZengMDPI AGSymmetry2073-89942022-09-011410197910.3390/sym14101979Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional GravityEmmanuele Battista0Giampiero Esposito1Department of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, AustriaDipartimento di Fisica “Ettore Pancini” and INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cintia Edificio 6, 80126 Napoli, ItalyThis paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a <i>n</i>-sided hyperbolic polygon <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">P</mi></semantics></math></inline-formula>, which is the Euclidean closure of the hyperbolic plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">H</mi></semantics></math></inline-formula>, bounded by <i>n</i> hyperbolic geodesic segments. The polygon <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">P</mi></semantics></math></inline-formula> is built by considering the unique geodesic that connects the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula> vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>z</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo>,</mo><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>z</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>z</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. The geodesics that link the vertices are Euclidean semicircles centred on the real axis. The vector normal to the geodesic linking two consecutive vertices is evaluated and turns out to be discontinuous. Within the framework of elliptic geometry, we solve the geodesic equation and construct a geodesic triangle. Additionally in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined.https://www.mdpi.com/2073-8994/14/10/1979geometric measure theorynon-euclidean geometriestwo-dimensional quantum gravity |
| spellingShingle | Emmanuele Battista Giampiero Esposito Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity Symmetry geometric measure theory non-euclidean geometries two-dimensional quantum gravity |
| title | Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity |
| title_full | Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity |
| title_fullStr | Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity |
| title_full_unstemmed | Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity |
| title_short | Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity |
| title_sort | discontinuous normals in non euclidean geometries and two dimensional gravity |
| topic | geometric measure theory non-euclidean geometries two-dimensional quantum gravity |
| url | https://www.mdpi.com/2073-8994/14/10/1979 |
| work_keys_str_mv | AT emmanuelebattista discontinuousnormalsinnoneuclideangeometriesandtwodimensionalgravity AT giampieroesposito discontinuousnormalsinnoneuclideangeometriesandtwodimensionalgravity |