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Parallel and totally geodesic hypersurfaces of 5-dimensional 2-step homogeneous nilmanifolds

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Abstract

In this paper we study parallel and totally geodesic hypersurfaces of two-step homogeneous nilmanifolds of dimension five. We give the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces. Moreover, we prove that two-step homogeneous nilmanifolds of dimension five which have one-dimensional centre never admit parallel hypersurfaces. Also we prove that the only two-step homogeneous nilmanifolds of dimension five which admit totally geodesic hypersurfaces have three-dimensional centre.

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References

  1. M. Aghasi, M. Nasehi: On homogeneous Randers spaces with Douglas or naturally reductive metrics. Differ. Geom. Dyn. Syst. 17 (2015), 1–12.

    MathSciNet  MATH  Google Scholar 

  2. M. Aghasi, M. Nasehi: On the geometrical properties of solvable Lie groups. Adv. Geom. 15 (2015), 507–517.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Aghasi, M. Nasehi: Some geometrical properties of a five-dimensional solvable Lie group. Differ. Geom. Dyn. Syst. 15 (2013), 1–12.

    MathSciNet  MATH  Google Scholar 

  4. M. Belkhelfa, F. Dillen, J. Inoguchi: Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces. PDEs, Submanifolds and Affine Differential Geometry, Warszawa, 2000 (B. Opozda, et al., eds.). Polish Academy of Sciences, Inst. Math., Warszawa, Banach Cent. Publ. 57, 2002, pp. 67–87.

    Chapter  Google Scholar 

  5. M. Božek: Existence of generalized symmetric Riemannian spaces with solvable isometry group. Čas. Pěst. Mat. 105 (1980), 368–384.

    MATH  Google Scholar 

  6. G. Calvaruso, O. Kowalski, R. A. Marinosci: Homogeneous geodesics in solvable Lie groups. Acta Math. Hungar. 101 (2003), 313–322.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Calvaruso, J. Van der Veken: Totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups. Results Math. 64 (2013), 135–153.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Calvaruso, J. Van der Veken: Parallel surfaces in three-dimensional Lorentzian Lie groups. Taiwanese J. Math. 14 (2010), 223–250.

    MathSciNet  MATH  Google Scholar 

  9. G. Calvaruso, J. Van der Veken: Lorentzian symmetric three-spaces and the classification of their parallel surfaces. Int. J. Math. 20 (2009), 1185–1205.

    Article  MathSciNet  MATH  Google Scholar 

  10. B.-Y. Chen: Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension. J. Geom. Phys. 60 (2010), 260–280.

    Article  MathSciNet  MATH  Google Scholar 

  11. B.-Y. Chen, J. Van der Veken: Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms. Tohoku Math. J. 61 (2009), 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. De Leo, J. Van der Veken: Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces. Geom. Dedicata 159 (2012), 373–387.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Homolya, O. Kowalski: Simply connected two-step homogeneous nilmanifolds of dimension 5. Note Mat. 26 (2006), 69–77.

    MathSciNet  MATH  Google Scholar 

  14. J. Inoguchi, J. Van der Veken: A complete classification of parallel surfaces in three-dimensional homogeneous spaces. Geom. Dedicata 131 (2008), 159–172.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Inoguchi, J. Van der Veken: Parallel surfaces in the motion groups E(1, 1) and E(2). Bull. Belg. Math. Soc.-Simon Stevin 14 (2007), 321–332.

    MathSciNet  MATH  Google Scholar 

  16. O. Kowalski: Generalized Symmetric Spaces. Lecture Notes in Mathematics 805, Springer, Berlin, 1980.

    Google Scholar 

  17. J. Lauret: Homogeneous nilmanifolds of dimension 3 and 4. Geom. Dedicata 68 (1997), 145–155.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. B. Lawson, Jr.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. (2) 89 (1969), 187–197.

    Article  MathSciNet  MATH  Google Scholar 

  19. H. R. Salimi Moghaddam: On the Randers metrics on two-step homogeneous nilmanifolds of dimension five. Int. J. Geom. Methods Mod. Phys. 8 (2011), 501–510.

    Article  MathSciNet  MATH  Google Scholar 

  20. U. Simon, A. Weinstein: Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie. Manuscr. Math. 1 (1969), 139–146. (In German.)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, Imam Khomeini Street, Isfahan, 84156-83111, Iran

    Mehri Nasehi

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  1. Mehri Nasehi
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Correspondence to Mehri Nasehi.

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Nasehi, M. Parallel and totally geodesic hypersurfaces of 5-dimensional 2-step homogeneous nilmanifolds. Czech Math J 66, 547–559 (2016). https://doi.org/10.1007/s10587-016-0274-x

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  • DOI: https://doi.org/10.1007/s10587-016-0274-x

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