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Abstract
We introduced special pseudo-Riemannian spaces, called geodesic A-symmetric spaces, into consideration. It is proven that there are no geodesic symmetric spaces and no geodesic Ricci symmetric spaces, which differ from spaces of constant curvature and Einstein spaces respectively. The research is carried out locally, by tensor methods, without any limitations imposed on a metric and a sign.
Keywords:
pseudo-Riemannian spaces, geodesic Ricci-symmetric spaces, geodesic mappings
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How to Cite
Kiosak, V., Kusik, L., & Isaiev, V. (2022). Geodesic Ricci-symmetric pseudo-Riemannian spaces. Proceedings of the International Geometry Center, 15(2), 110-120. https://doi.org/10.15673/tmgc.v15i2.2224
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References
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10. V. Kiosak, O. Prishlyak, and O. Lesechko. On the geodesic mappings of pseudo-Riemannian spaces with special supplementary tensor. Proceedings of the International Geometry Center, 14(4):13-26, 2021. https://doi.org/10.15673/tmgc.v14i4.2140 pathdoi:10.15673/tmgc.v14i4.2140.
11. V. Kiosak, A. Savchenko, and A. Kamienieva. Geodesic mappings of compact quasi-Einstein spaces with constant scalar curvature. AIP Conference Proceedings, 2302(040002), 2020. https://doi.org/10.1063/5.0033661 pathdoi:10.1063/5.0033661.
12. V. Kiosak, A. Savchenko, and S. Khniunin. On the typology of quasi-Einstein spaces. AIP Conference Proceedings, 2302(040003), 2020. https://doi.org/10.1063/5.0033700 pathdoi:10.1063/5.0033700.
13. V. Kiosak, A. Savchenko, and G. Kovalova. Geodesic mappings of compact quasi-Einstein spaces, l. Proceedings of the International Geometry Center, 13(1):35-48, 2020. https://doi.org/10.15673/tmgc.v13i1.1711 pathdoi:10.15673/tmgc.v13i1.1711.
14. V. Kiosak, A. Savchenko, and O. Latysh. Geodesic mappings of compact quasi-Einstein spaces, II. Proceedings of the International Geometry Center, 14(1):80-91, 2021. https://doi.org/10.15673/tmgc.v14i1.1936 pathdoi:10.15673/tmgc.v14i1.1936.
15. G. I. Kruchkovich. Riemannian and pseudo-Riemannian spaces. Itogi Nauki. Ser. Mat. Algebra. Topol. Geom., pages 191-220, 1968.
16. T. Levi-Civita. Sulle transformationi delle equazioni dinamiche. Ann. Mat. Milano, Ser. 2., 24:255-300, 1896. https://doi.org/10.1007/bf02419530 pathdoi:10.1007/bf02419530.
17. J. Mikes. Geodesic mappings of Einstein spaces. Math. Notes, 28:922-923, 1981.
18. J. Mikes, I. Hinterleitner, and V. Kiosak. On the theory of geodesic mappings of Einstein spaces and their generalizations. AIP Conference Proceedings, 861:428-435, 2006. https://doi.org/10.1063/1.2399606 pathdoi:10.1063/1.2399606.
19. J. Mikes, V. Kiosak, and O. Vanzurova. Geodesic mappings of manifolds with affine connection. Palacky University Press, Olomouc, 2008.
20. N. S. Sinyukov. Geodesic mappings of Riemannian spaces. Nauka, 1979.
21. V. S. Sobchuk. Riemannian spaces which admit a generalized-recurrent symmetric tensor of the second order. Dokl. Akad. Nauk SSSR, 185(6):1247-1250, 1969.
22. V. S. Sobchuk. Ricci generalized symmetric Riemannian spaces admit nontrivial geodesic mappings. Dokl. Akad. Nauk SSSR, 267(4):793-795, 1982.
23. V. S. Sobchuk. Geodesic mappings of some classes of Riemannian spaces. Soviet Math. (Iz. VUZ), 34(4):56-59, 1990.
24. V. S. Sobchuk. Geodesic mapping of Ricci 4-symmetric Riemannian spaces. Soviet Math. (Iz. VUZ), 35(4):68-69, 1991.
25. A. S. Solodovnikov. Geodesic classes of V(K) spaces. Dokl. Akad. Nauk SSSR, 141:322-325, 1956.
26. A. S. Solodovnikov. Geometric description of all possible representations of a Riemannian metric in Levi-Cività form. Dokl. Akad. Nauk SSSR, 111:33-36, 1961.
27. H. Weyl. Zur infinitesimal geometrie Einordnung der projectiven und der konformen Auffassung. Gottinger Nachtr, pages 99-112, 1921.
2. L. P. Eisenhart. Riemannian geometry. Princeton University Press, 1997.
3. I. Hinterleitner and V. Kiosak. Special Einstein's equations on Kahler manifolds. Archivum Mathematicum, 46(5):333-337, 2010.
4. V. F. Kagan. Subprojective spaces. Moscow:Fizmatgiz, 1961.
5. V. Kiosak and G. Kovalova. Geodesic mappings of quasi-Einstein spaces with a constant scalar curvature. Matematychni Studii, 53(2):212-217, 2020. https://doi.org/10.30970/ms.53.2.212-217 pathdoi:10.30970/ms.53.2.212-217.
6. V. Kiosak and V. Matveev. Complete Einstein metrics are geodesically rigid. Communications in Mathematical Physics, 289(1):383-400, 2009. https://doi.org/10.1007/s00220-008-0719-7 pathdoi:10.1007/s00220-008-0719-7.
7. V. Kiosak and V. Matveev. Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two. Communications in Mathematical Physics, 297(2):401-426, 2010. https://doi.org/10.1007/s00220-010-1037-4 pathdoi:10.1007/s00220-010-1037-4.
8. V. Kiosak and V. Matveev. There exist no 4-dimensional geodesically equivalent metrics with the same stress-energy tensor. Journal of Geometry and Physics, 78:1-11, 2014. https://doi.org/10.1016/j.geomphys.2014.01.002 pathdoi:10.1016/j.geomphys.2014.01.002.
9. V. Kiosak, V. Matveev, J. Mikes, and I. Shandra. On the degree of geodesic mobility for Riemannian metrics. Mathematical Notes, 87(3-4):586-587, 2010. https://doi.org/10.1134/S0001434610030375 pathdoi:10.1134/S0001434610030375.
10. V. Kiosak, O. Prishlyak, and O. Lesechko. On the geodesic mappings of pseudo-Riemannian spaces with special supplementary tensor. Proceedings of the International Geometry Center, 14(4):13-26, 2021. https://doi.org/10.15673/tmgc.v14i4.2140 pathdoi:10.15673/tmgc.v14i4.2140.
11. V. Kiosak, A. Savchenko, and A. Kamienieva. Geodesic mappings of compact quasi-Einstein spaces with constant scalar curvature. AIP Conference Proceedings, 2302(040002), 2020. https://doi.org/10.1063/5.0033661 pathdoi:10.1063/5.0033661.
12. V. Kiosak, A. Savchenko, and S. Khniunin. On the typology of quasi-Einstein spaces. AIP Conference Proceedings, 2302(040003), 2020. https://doi.org/10.1063/5.0033700 pathdoi:10.1063/5.0033700.
13. V. Kiosak, A. Savchenko, and G. Kovalova. Geodesic mappings of compact quasi-Einstein spaces, l. Proceedings of the International Geometry Center, 13(1):35-48, 2020. https://doi.org/10.15673/tmgc.v13i1.1711 pathdoi:10.15673/tmgc.v13i1.1711.
14. V. Kiosak, A. Savchenko, and O. Latysh. Geodesic mappings of compact quasi-Einstein spaces, II. Proceedings of the International Geometry Center, 14(1):80-91, 2021. https://doi.org/10.15673/tmgc.v14i1.1936 pathdoi:10.15673/tmgc.v14i1.1936.
15. G. I. Kruchkovich. Riemannian and pseudo-Riemannian spaces. Itogi Nauki. Ser. Mat. Algebra. Topol. Geom., pages 191-220, 1968.
16. T. Levi-Civita. Sulle transformationi delle equazioni dinamiche. Ann. Mat. Milano, Ser. 2., 24:255-300, 1896. https://doi.org/10.1007/bf02419530 pathdoi:10.1007/bf02419530.
17. J. Mikes. Geodesic mappings of Einstein spaces. Math. Notes, 28:922-923, 1981.
18. J. Mikes, I. Hinterleitner, and V. Kiosak. On the theory of geodesic mappings of Einstein spaces and their generalizations. AIP Conference Proceedings, 861:428-435, 2006. https://doi.org/10.1063/1.2399606 pathdoi:10.1063/1.2399606.
19. J. Mikes, V. Kiosak, and O. Vanzurova. Geodesic mappings of manifolds with affine connection. Palacky University Press, Olomouc, 2008.
20. N. S. Sinyukov. Geodesic mappings of Riemannian spaces. Nauka, 1979.
21. V. S. Sobchuk. Riemannian spaces which admit a generalized-recurrent symmetric tensor of the second order. Dokl. Akad. Nauk SSSR, 185(6):1247-1250, 1969.
22. V. S. Sobchuk. Ricci generalized symmetric Riemannian spaces admit nontrivial geodesic mappings. Dokl. Akad. Nauk SSSR, 267(4):793-795, 1982.
23. V. S. Sobchuk. Geodesic mappings of some classes of Riemannian spaces. Soviet Math. (Iz. VUZ), 34(4):56-59, 1990.
24. V. S. Sobchuk. Geodesic mapping of Ricci 4-symmetric Riemannian spaces. Soviet Math. (Iz. VUZ), 35(4):68-69, 1991.
25. A. S. Solodovnikov. Geodesic classes of V(K) spaces. Dokl. Akad. Nauk SSSR, 141:322-325, 1956.
26. A. S. Solodovnikov. Geometric description of all possible representations of a Riemannian metric in Levi-Cività form. Dokl. Akad. Nauk SSSR, 111:33-36, 1961.
27. H. Weyl. Zur infinitesimal geometrie Einordnung der projectiven und der konformen Auffassung. Gottinger Nachtr, pages 99-112, 1921.