Proceedings of the International Geometry Center

ISSN-print: 2072-9812
ISSN-online: 2409-8906
ISO: 26324:2012
Archives

Fundamental theorems of quasi-geodesic mappings of generalized-recurrent-parabolic spaces

##plugins.themes.bootstrap3.article.sidebar##

pdf
Published Nov 12, 2023
DOI https://doi.org/10.15673/pigc.v16i3.2576

##plugins.themes.bootstrap3.article.main##

Irina Kurbatova
Odesa National University of Technology
https://orcid.org/0000-0003-0215-6060
Margaret Pistruil
Odesa National University of Technology
Nadiia Konovenko
Odesa National University of Technology
https://orcid.org/0000-0002-8631-0688

Abstract

In previous papers we studied mappings of pseudo-Riemannian spaces being mutually quasi-geodesic and almost geodesic of the 2nd type. As a result, we arrived at the quasi-geodesic mapping f: (Vn, gij, Fih) → (Vn, gij, Fih) of spaces with an affine structure, which was called generalized-recurrent. Quasi-geodesic mappings are divided into two types: general and canonical. In this article, the fundamental issues of the theory of quasi-geodesic mappings of generalized-recurrent-parabolic spaces are considered. First, the fundamental equations of quasi-geodesic mappings are reduced to a form that allows effective investigation. Then, using a new form of the fundamental equations, we prove theorems that allow for any generalized-recurrent-parabolic space (Vn, gij, Fih) or to find all spaces (Vn, gij, Fih) onto which Vn admits a quasi-geodesic mapping of the general form, or prove that there are no such spaces.

Keywords:
affine structure, quasi-geodesic mapping, pseudo-Riemannian space

##plugins.themes.bootstrap3.article.details##

How to Cite
Kurbatova, I., Pistruil, M., & Konovenko, N. (2023). Fundamental theorems of quasi-geodesic mappings of generalized-recurrent-parabolic spaces. Proceedings of the International Geometry Center, 16(3-4), 217-230. https://doi.org/10.15673/pigc.v16i3.2576
Issue
Vol 16 No 3-4 (2023)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgment of the work’s authorship and initial publication in this journal.

Provided they are the owners of the copyright to their work, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository, in a journal or publish it in a book), with an acknowledgment of its initial publication in this journal.

Authors are permitted and encouraged to post their work online (e.g., in institutional repositories, disciplinary repositories, or on their website) prior to and during the submission process.

Author Biographies

Irina Kurbatova, Odesa National University of Technology

Odesa National University of Technology

Margaret Pistruil, Odesa National University of Technology

Odesa National University of Technology

Nadiia Konovenko, Odesa National University of Technology

Odesa National University of Technology

References

1. D. V. Beklemivsev. Differential geometry of spaces with almost complex structure. In Geometry 1963 (Russian), Itogi Nauki, pages 165-212. Akad. Nauk SSSR Inst. Nauvcn. Informacii, Moscow, 1965.
2. V. E. Berezovskii and J. Mikesh. Almost geodesic mappings of spaces with affine connection. J. Math. Sci. (N.Y.), 207(3):389-409, 2015. Translated from Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz. Vol. 126, Geometry, 2013. https://doi.org/10.1007/s10958-015-2378-5 pathdoi:10.1007/s10958-015-2378-5.
3. V. E. Berezovskiui, uI. Mikesh, G. Khuda, and E. E. Chepurnaya. Canonical almost geodesic mappings that preserve the projective curvature tensor. Izv. Vyssh. Uchebn. Zaved. Mat., (6):3-8, 2017. https://doi.org/10.3103/s1066369x17060019 pathdoi:10.3103/s1066369x17060019.
4. H. Chuda, J. Mikesh, P. Pevska, and M. Shiha. On holomorphically projective mappings of equidistant parabolic Kahler spaces. In Geometry, integrability and quantization XIX, pages 115-121. Bulgar. Acad. Sci., Sofia, 2018.
5. D. Doikov and V. Kiosak. On the Schwarzschild model for gravitating objects of the universe. AIP Conference Proceedings, 2302:040001, 2020.
6. I. Hinterleitner and V. Kiosak. Special Einstein's equations on Kahler manifolds. Arch. Math. (Brno), 46(5):333-337, 2010. https://doi.org/10.5817/am2012-5-333 pathdoi:10.5817/am2012-5-333.
7. I. Hinterleitner, J. Mikesh, and P. Pevska. Fundamental equations of F-planar mappings. Lobachevskii J. Math., 38(4):653-659, 2017. https://doi.org/10.1134/S1995080217040096 pathdoi:10.1134/S1995080217040096.
8. V. Kiosak, A. Savchenko, and A. Kamienieva. Geodesic mappings of compact quasi-Einstein spaces with constant scalar curvature. AIP Conference Proceedings 2302, page 040002, 2020.
9. V. Kiosak, A. Savchenko, and S. Khniunin. On the typology of quasi-Einstein spaces. AIP Conference Proceedings 2302, page 040003, 2020.
10. V. Kiosak, A. Savchenko, and O. Latysh. Geodesic mappings of compact quasi-Einstein spaces, II. Proc. Int. Geom. Cent., 14(1):80-91, 2021. https://doi.org/10.15673/tmgc.v14i1.1936 pathdoi:10.15673/tmgc.v14i1.1936.
11. V. Kiosak, A. Savchenko, and T. Shevchenko. Holomorphically projective mappings of special Kähler manifolds. AIP Conference Proceedings, 2025( 080004), 2018.
12. I. Kurbatova. Canonical quasi-geodesic mappings of Kӓhler spaces. Proc. Intern. Geom. Center, 7(1):53-64, 2014.
13. I. Kurbatova. On laws of canonical quasi-geodesic mappings of parabolically Kӓhler spaces. Proc. Intern. Geom. Center, 7(2):26-35, 2014.
14. I. Kurbatova and D. Lozienko. On canonical quasigeodesic mappings of recurrent parabolic spaces. Proc. Intern. Geom. Center, 10(3-4):44-57, 2017.
15. I. Kurbatova and M. Pistruil. Quasigeodesic mappings of special pseudo-Riemannian spaces. Proc. Int. Geom. Cent., 13(3):18-32, 2020. https://doi.org/10.15673/tmgc.v13i3.1770 pathdoi:10.15673/tmgc.v13i3.1770.
16. I. Kurbatova and O. Sisyuk. Quasi-geodesic mappings of recurrent parabolically spaces. Proc. Intern. Geom. Center, 8(1):74-83, 2014.
17. J. Mikesh. Holomorphically projective mappings and their generalizations. volume 89, pages 1334-1353. 1998. Geometry, 3. https://doi.org/10.1007/BF02414875 pathdoi:10.1007/BF02414875.
18. J. Mikesh and N. S. Sinyukov. Quasiplanar mappings of spaces with affine connection. Izv. Vyssh. Uchebn. Zaved. Mat., (1):55-61, 1983.
19. J. Mikesh, A. Vanvzurova, and I. Hinterleitner. Geodesic mappings and some generalizations. Palacky University Olomouc, Faculty of Science, Olomouc, 2009.
20. A. Z. Petrov. Modeling of the paths of test particles in gravitation theory. Gravitacija i Teor. Otnositelcprime nosti, (6):7-21, 1969.
21. Milovs Z. Petrovic and P. Pevska. Equitorsion holomorphically projective mappings of generalized m-parabolic Kahler manifolds. Filomat, 33(4):1047-1052, 2019. https://doi.org/10.2298/fil1904047p pathdoi:10.2298/fil1904047p.
22. P. Pevska, J. Mikesh, H. Chuda, and M. Shiha. On holomorphically projective mappings of parabolic Kahler manifolds. Miskolc Math. Notes, 17(2):1011-1019, 2016. https://doi.org/10.18514/MMN.2017.1893 pathdoi:10.18514/MMN.2017.1893.
23. M. I. Pistruil and I. M. Kurbatova. Canonical quasi-geodesic mappings of special pseudo-Riemannian spaces. Proc. Int. Geom. Cent., 15(3-4):163-176, 2022.
24. A. Savchenko, N. Vashpanova, and N. Vasylieva. Generalized varphi(Ric)-vector fields in special pseudo-Riemannian spaces. Proc. Int. Geom. Cent., 14(4):231-242, 2021. https://doi.org/10.15673/tmgc.v14i4.2155 pathdoi:10.15673/tmgc.v14i4.2155.
25. M. Shiha. Geodesic and holomorphically projective mappings of parabolically Kahlerian spaces. PhD thesis, Odessa: Univ., Supervisor Mikeš, J., 1992.
26. N. S. Sinjukov. cyr Geodezicheskie otobrazheniya rimanovykh prostranstv. \"Nauka\", Moscow, 1979.
27. N. S. Sinjukov. cyr Geodezicheskie otobrazheniya rimanovykh prostranstv. \"Nauka\", Moscow, 1979.
28. N. S. Sinyukov. Almost geodesic mappings of affinely connected and Riemannian spaces. In Problems in geometry, Vol. 13, Itogi Nauki i Tekhniki, pages 3-26, 199. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1982.
29. M. S. Stankovic, M. L. Zlatanovic, and N. O. Vesic. Basic equations of G-almost geodesic mappings of the second type, which have the property of reciprocity. Czechoslovak Math. J., 65(140)(3):787-799, 2015. https://doi.org/10.1007/s10587-015-0208-z pathdoi:10.1007/s10587-015-0208-z