Skip to main content
Log in

Second-Order Parallel Tensors on 𝒮-Manifolds and Semiparallel Hypersurfaces of 𝒮-Space Forms

  • BRIEF COMMUNICATIONS
  • Published:
Ukrainian Mathematical Journal Aims and scope

We study a second-order parallel symmetric tensor in an 𝒮-manifold and show that there is no semiparallel hypersurface in the 𝒮-space forms \( {\tilde{M}}^{2n+s}(c) \) with cs.

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.

References

  1. A. C. Asperti, G. A. Lobos, and F. Mercuri, “Pseudo-parallel submanifolds of a space form,” Adv. Geom., 2, 57–71 (2002).

    MathSciNet  MATH  Google Scholar 

  2. D. E. Blair, “Contact manifolds in Riemannian geometry,” Lect. Notes Math., 509 (1976).

  3. D. E. Blair, “Geometry of manifolds with structural group 𝒰(n) ∗ 𝒪(s),” J. Different. Geom., 4, 155–167 (1970).

    Article  MathSciNet  Google Scholar 

  4. J. L. Cabrerizo, L. M. Fernandez, and M. Fernandez, “The curvature of submanifolds of 𝒮-space forms,” Acta Math. Hungar., 62, No. 3-4, 373–383 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. L. Cabrerizo, L. M. Fernandez, and M. Fernandez, “On pseudo-umbilical hypersurfaces of S-manifolds,” Acta Math. Hungar., 70, No. 1-2, 121–128 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Y. Chen, Geometry of Submanifolds and Its Applications, Science University of Tokyo, Japon (1981).

    MATH  Google Scholar 

  7. J. Deprez, “Semiparallel surfaces in Euclidean space,” J. Geom., 25, 192–200 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Deprez, “Semiparallel hypersurfaces,” Rend. Semin. Mat. Univ. Politec. Torino, 44, 303–316 (1986).

    MathSciNet  MATH  Google Scholar 

  9. R. Deszcz, “On pseudosymmetric spaces,” Bull. Soc. Math. Belg. Ser. A., 44, 1–34 (1992).

    MathSciNet  MATH  Google Scholar 

  10. F. Dillen, “Semiparallel hypersurfaces of real space forms,” Israel J. Math., 75, 193–202 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  11. L. P. Eisenhart, “Symmetric tensors of the second order whose first covariant derivates are zero,” Trans. Amer. Math. Soc., 25, 297–306 (1923).

    Article  MathSciNet  Google Scholar 

  12. F. Gherib and M. Belkhelfa, “Parallel submanifolds of generalized Sasakian space forms,” Bull. Transilv. Univ. Braşov Ser. III, 2, 51 (2009).

  13. F. Gherib and M. Belkhelfa, “Second order parallel tensors on generalized Sasakian space forms and semiparallel hypersurfaces in Sasakian space forms,” Beitr. Algebra Geom., 51, No. 1, 1–7 (2010).

    MATH  Google Scholar 

  14. I. Hasegawa, Y. Okuyama, and T. Abe, “On p-th Sasakian manifolds,” J. Hokkaido Univ. Educ., 37, No. 1, 1–16 (1986).

    MathSciNet  Google Scholar 

  15. R. Kaid and M. Belkhelfa, “Symmetry properties of 𝒮-space forms,” J. Geom., 106, No. 3, 513–530 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Kobayashi and S. Tsuchiya, “Invariant submanifolds of f-manifolds with complemented frames,” Kodai Math. Sem. Rep., 24, 430–450 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Levy, “Symmetric tensors of the second order whose covariant derivates vanish,” Ann. Math., 27, 91–98 (1926).

    Article  MATH  Google Scholar 

  18. M. M. Tripathi and K. D. Singh, “On submanifolds of 𝒮-manifolds,” Ganita, 47, No. 2, 51–54 (1996).

  19. K. Yano, “On a structure defined a tensor field f of type (1, 1) satisfying f3 + f = 0,” Tensor (N.S.), 14, 99–109 (1963).

  20. K. Yano and M. Kon, “Structures on manifolds,” Ser. Pure Math., 13 (1984).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Belkhelfa.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 10, pp. 1422–1429, October, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belkhelfa, M., Mahi, F. Second-Order Parallel Tensors on 𝒮-Manifolds and Semiparallel Hypersurfaces of 𝒮-Space Forms. Ukr Math J 71, 1627–1635 (2020). https://doi.org/10.1007/s11253-020-01735-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-020-01735-8

Navigation