Skip to main content
Log in

Symmetric, Semisymmetric, and Recurrent Projectively Euclidean Spaces

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we present some results obtained for symmetric, semisymmetric, and semisymmetric recurrent projectively Euclidean spaces. Components of objects of affine connections of these spaces are found.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. V. E. Berezovskii, N. I. Guseva, and J. Mikeš, “On a particular case of first-type almost geodesic mappings of affinely connected spaces that preserve a certain tensor,” Mat. Zametki, 98, No. 3, 463–466 (2015).

    MathSciNet  Google Scholar 

  2. V. Berezovskii, I. Hinterleitner, and J. Mikeš, “Geodesic mappings of manifolds with affine connection onto the Ricci symmetric manifolds,” Filomat, 32, No. 2, 379–385 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. E. Berezovskii, L. E. Kovalev, and J. Mikeš, “On preserving of the Riemann tensor under some mappings of affinely connected spaces,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 9, 3-10 (2018).

  4. É. Cartan, “Sur les variété à connexion projective,” Bull. Soc. Math. Fr., 52, 205–241 (1924).

    Article  MATH  Google Scholar 

  5. É. Cartan, “Sur une classe remarquable d’espaces de Riemann, II,” Bull. Soc. Math. Fr., 55, 114–134 (1927).

    Article  MathSciNet  Google Scholar 

  6. L. P. Eisenhart, Non-Riemannian geometry, Princeton Univ. Press (1926).

    MATH  Google Scholar 

  7. I. Hinterleitner and J. Mikeš, “Geodesic mappings onto Weyl manifolds,” J. Appl. Math., 2, No. 1, 125–133 (2009).

    MathSciNet  Google Scholar 

  8. I. Hinterleiner and J. Mikeš, “Fundamental equations of geodesic mappings and their generalizations,” J. Math. Sci., 174, 537–554 (2011).

    Article  MATH  Google Scholar 

  9. I. Hinterleiner and J. Mikeš, “Projective equivalence and spaces with equiaffine connection,” J. Math. Sci., 177, 546–550 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. I. Hinterleiner and J. Mikeš, “Geodesic mappings and Einstein spaces,” in: Geometric Methods in Physics, Birkhäuser, Basel (2013), pp. 331–335.

  11. I. Hinterleiner and J. Mikeš, “Geodesic Mappings of (pseudo-) Riemannian manifolds preserve class of differentiability,” Miskolc. Math. Notes., 14, 89–96 (2013).

    MathSciNet  Google Scholar 

  12. I. Hinterleiner and J. Mikeš, “Geodesic mappings and differentiability of metrics, affine and projective connections,” Filomat, 29, 1245–1249 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Hinterleiner and J.Mikeš, “Geodesic mappings onto Riemannian manifolds and differentiability,” in: Geometry, Integrability and Quantization. Vol. 18, Bulgar. Acad. Sci., Sofia (2017), pp. 183–190.

  14. V. R. Kaigorodov, “Structure of the curvature of the space-time,” Itogi Nauki Tekhn. Ser. Probl. Geom., 14, 177-204 (1983).

    MathSciNet  MATH  Google Scholar 

  15. P. I. Kovalev, “Triple Lie systems and spaces of affine connection,” Mat. Zametki, 14, No. 1, 107-112 (1973).

    MathSciNet  Google Scholar 

  16. A. Lichnerowicz, “Courbure, nombres de Betti, et espaces symétriques,” in: Proc. Int. Congr. Math. (Cambridge, Massachusetts, August 30–September 6, 1950), Am. Math. Soc. (1952), pp. 216–223.

  17. Yu. G. Lumiste, “Semisymmetric submanifolds,” J. Math. Sci., 70, No. 2 (1994), pp. 1609–1623.

    Article  MATH  Google Scholar 

  18. J. Mikeš, Geodesic mappings of semisymmetric spaces, preprint VINITI No. 3924-76Dep (1976).

  19. J. Mikeš, “Geodesic mappings onto semisymmetric space,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 2 (1994), pp. 37-43.

  20. J. Mikeš, “Geodesic mappings of affine-connected and Riemannian spaces,” J. Math. Sci., 78 (1996), pp. 311–333.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Mikeš, V. E. Berezovskii, E. Stepanova, and H. Chudá, “Geodesic mappings and their generalizations,” J. Math. Sci., 217 (2016), pp. 607–623.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Mikeš and M. Chodorová, “On concircular and torse-forming vector fields on compact manifolds,” Acta Math. Acad. Paedagog. Nyházi., 26 (2010), pp. 329–335.

    MathSciNet  MATH  Google Scholar 

  23. J. Mikeš, A. Vanžurová, and I. Hinterleitner, Geodesic Mappings and Some Generalizations, Palacky Univ. Press, Olomouc (2009).

    MATH  Google Scholar 

  24. J. Mikeš et al., Differential Geometry of Special Mappings, Palacky Univ. Press, Olomouc (2015).

    MATH  Google Scholar 

  25. V. A. Mirzoyan, “Classification of Ric-semiparallel hypersurfaces in Euclidean spaces,” Mat. Sb., 191, No. 9 (2000), pp. 65-80.

    MathSciNet  MATH  Google Scholar 

  26. M. Najdanović, M. Zlatanović, and I. Hinterleitner, “Conformal and geodesic mappings of generalized equidistant spaces,” Publ. Inst. Math. Beograd., 98 (112) (2015), pp. 71–84.

    Article  MathSciNet  MATH  Google Scholar 

  27. K. Nomizu, “On hypersurfaces satisfying a certain condition on the curvature tensor,” Tôhoku Math. J., 20 (1968), pp. 46–59.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. P. Norden, Spaces of Affine Connection [in Russian], Nauka, Moscow (1976).

    MATH  Google Scholar 

  29. A. Z. Petrov, New Methods in General Relativity [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  30. H. S. Ruse, “On simply harmonic spaces,” J. London Math. Soc., 21 (1947), pp. 243–247.

    MathSciNet  MATH  Google Scholar 

  31. A. A. Sabykanov, J. Mikeš, and P. Peška, “On semisymmetric projectively Euclidean spaces,” in: Proc Int. Conf. “Modern Geometry and Its Applications” (Kazan, Novermber 27 – December 3, 2017), Kazan Federal Univ., Kazan (2017), pp. 123–125.

  32. A. Sabykanov, J.Mikeš, and P. Peška, “Recurrent equiaffine projective-Euclidean spaces,” Filomat, 33, No. 4 (2019).

  33. K. Sekigawa, “On some hypersurfaces satisfying R(X, Y )∙R = 0,” Tensor, 25 (1972), pp. 133–136.

    MathSciNet  MATH  Google Scholar 

  34. Shirokov P. A., “Constant fields of vectors and tensors in Riemannian spaces,” Izv. Kazan. Fiz.- Mat. Obshch., No. 25 (1925), pp. 86–114.

  35. Shirokov P. A., Selected Works in Geometry [in Russian], Kazan (1966).

  36. N. S. Sinyukov, “On geodesic mappings of Riemannian spaces on symmetric Riemannian spaces,” Dokl. Akad. Nauk SSSR, 98, No. 1 (1954), pp. 21–23.

    MathSciNet  MATH  Google Scholar 

  37. N. S. Sinyukov, Geodesic Mappings of Riemannian spaces [in Russian], Nauka, Moscow (1979).

    MATH  Google Scholar 

  38. N. S. Sinyukov, “Almost geodesic mappings of affinely connected and Riemannian spaces,” Itogi Nauki Tekhn. Ser. Probl. Geom., 13 (1982), pp. 3-26.

    MathSciNet  MATH  Google Scholar 

  39. M. Stanković, S. Mincić, Lj. Velimirović, and M. Zlatanović, “On equitorsion geodesic mappings of general affine connection spaces,” Rend. Semin. Mat. Univ. Padova, 124 (2010), pp. 77–90.

  40. S. Stepanov, I. Shandra, and J. Mikeš, “Harmonic and projective diffeomorphisms,” J. Math. Sci., 207 (2015), pp. 658–668.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. S. Sabykanov.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 179, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 1, 2020.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sabykanov, A.S., Mikeš, J. & Peška, P. Symmetric, Semisymmetric, and Recurrent Projectively Euclidean Spaces. J Math Sci 276, 410–416 (2023). https://doi.org/10.1007/s10958-023-06757-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06757-8

Keywords and phrases

AMS Subject Classification

Navigation