Topological structure of optimal flows on the Girl's surface
We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Odesa National University of Technology
2023-02-01
|
| Series: | Pracì Mìžnarodnogo Geometričnogo Centru |
| Subjects: | |
| Online Access: | https://journals.ontu.edu.ua/index.php/geometry/article/view/2338 |
| _version_ | 1811159796240351232 |
|---|---|
| author | Alexandr Prishlyak Maria Loseva |
| author_facet | Alexandr Prishlyak Maria Loseva |
| author_sort | Alexandr Prishlyak |
| collection | DOAJ |
| description | We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a $2$-disk, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the $1$-skeleton of the surface. Second, we describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows. |
| first_indexed | 2024-04-10T05:48:21Z |
| format | Article |
| id | doaj.art-29e8eebb46934506b96514808cfc8000 |
| institution | Directory Open Access Journal |
| issn | 2072-9812 2409-8906 |
| language | English |
| last_indexed | 2024-04-10T05:48:21Z |
| publishDate | 2023-02-01 |
| publisher | Odesa National University of Technology |
| record_format | Article |
| series | Pracì Mìžnarodnogo Geometričnogo Centru |
| spelling | doaj.art-29e8eebb46934506b96514808cfc80002023-03-05T07:59:14ZengOdesa National University of TechnologyPracì Mìžnarodnogo Geometričnogo Centru2072-98122409-89062023-02-01153-418420210.15673/tmgc.v15i3-4.23382338Topological structure of optimal flows on the Girl's surfaceAlexandr Prishlyak0Maria Loseva1Taras Shevchenko National University of KyivTaras Shevchenko National University of KyivWe investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a $2$-disk, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the $1$-skeleton of the surface. Second, we describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows.https://journals.ontu.edu.ua/index.php/geometry/article/view/2338morse-smale flowtopological equivalenceprojective plane |
| spellingShingle | Alexandr Prishlyak Maria Loseva Topological structure of optimal flows on the Girl's surface Pracì Mìžnarodnogo Geometričnogo Centru morse-smale flow topological equivalence projective plane |
| title | Topological structure of optimal flows on the Girl's surface |
| title_full | Topological structure of optimal flows on the Girl's surface |
| title_fullStr | Topological structure of optimal flows on the Girl's surface |
| title_full_unstemmed | Topological structure of optimal flows on the Girl's surface |
| title_short | Topological structure of optimal flows on the Girl's surface |
| title_sort | topological structure of optimal flows on the girl s surface |
| topic | morse-smale flow topological equivalence projective plane |
| url | https://journals.ontu.edu.ua/index.php/geometry/article/view/2338 |
| work_keys_str_mv | AT alexandrprishlyak topologicalstructureofoptimalflowsonthegirlssurface AT marialoseva topologicalstructureofoptimalflowsonthegirlssurface |