On the geometrical properties of Heisenberg groups

Mehri Nasehi

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 1, page 11-19
  • ISSN: 0044-8753

Abstract

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In [20] the existence of major differences about totally geodesic two-dimensional foliations between Riemannian and Lorentzian geometry of the Heisenberg group is proved. Our aim in this paper is to obtain a comparison on some other geometrical properties of these spaces. Interesting behaviours are found. Also the non-existence of left-invariant Ricci and Yamabe solitons and the existence of algebraic Ricci soliton in both Riemannian and Lorentzian cases are proved. Moreover, all of the descriptions of their homogeneous Riemannian and Lorentzian structures and their types are obtained. Besides, all the left-invariant generalized Ricci solitons and unit time-like vector fields which are spatially harmonic are completely determined.

How to cite

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Nasehi, Mehri. "On the geometrical properties of Heisenberg groups." Archivum Mathematicum 056.1 (2020): 11-19. <http://eudml.org/doc/295087>.

@article{Nasehi2020,
abstract = {In [20] the existence of major differences about totally geodesic two-dimensional foliations between Riemannian and Lorentzian geometry of the Heisenberg group $H_\{3\}$ is proved. Our aim in this paper is to obtain a comparison on some other geometrical properties of these spaces. Interesting behaviours are found. Also the non-existence of left-invariant Ricci and Yamabe solitons and the existence of algebraic Ricci soliton in both Riemannian and Lorentzian cases are proved. Moreover, all of the descriptions of their homogeneous Riemannian and Lorentzian structures and their types are obtained. Besides, all the left-invariant generalized Ricci solitons and unit time-like vector fields which are spatially harmonic are completely determined.},
author = {Nasehi, Mehri},
journal = {Archivum Mathematicum},
keywords = {left-invariant generalized Ricci solitons; harmonicity of invariant vector fields; homogeneous structures},
language = {eng},
number = {1},
pages = {11-19},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the geometrical properties of Heisenberg groups},
url = {http://eudml.org/doc/295087},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Nasehi, Mehri
TI - On the geometrical properties of Heisenberg groups
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 1
SP - 11
EP - 19
AB - In [20] the existence of major differences about totally geodesic two-dimensional foliations between Riemannian and Lorentzian geometry of the Heisenberg group $H_{3}$ is proved. Our aim in this paper is to obtain a comparison on some other geometrical properties of these spaces. Interesting behaviours are found. Also the non-existence of left-invariant Ricci and Yamabe solitons and the existence of algebraic Ricci soliton in both Riemannian and Lorentzian cases are proved. Moreover, all of the descriptions of their homogeneous Riemannian and Lorentzian structures and their types are obtained. Besides, all the left-invariant generalized Ricci solitons and unit time-like vector fields which are spatially harmonic are completely determined.
LA - eng
KW - left-invariant generalized Ricci solitons; harmonicity of invariant vector fields; homogeneous structures
UR - http://eudml.org/doc/295087
ER -

References

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  1. Batat, W., Onda, K., 10.1016/j.geomphys.2016.11.018, J. Geom. Phys. 114 (2017), 138–152. (2017) MR3610038DOI10.1016/j.geomphys.2016.11.018
  2. Batat, W., Rahmani, S., Homogeneous Lorentzian structures on the generalized Heisenberg group, Differ. Geom. Dyn. Syst. 12 (2010), 12–17. (2010) MR2606543
  3. Brozos-Vazquez, M., Calvaruso, G., Garcia-Rio, E., Gavino-Fernandez, S., 10.1007/s11856-011-0124-3, Israel J. Math. 188 (2012), 385–403. (2012) MR2897737DOI10.1007/s11856-011-0124-3
  4. Calvaruso, G., 10.1007/s10711-007-9163-7, Geom. Dedicata 127 (2007), 99–119. (2007) MR2338519DOI10.1007/s10711-007-9163-7
  5. Calvaruso, G., 10.1016/j.geomphys.2010.11.001, J. Geom. Phys. 61 (2011), 498–515. (2011) MR2746133DOI10.1016/j.geomphys.2010.11.001
  6. Calvaruso, G., 10.2478/s11533-011-0109-9, Cent. Eur. J. Math. 10 (2) (2012), 411–425. (2012) MR2886549DOI10.2478/s11533-011-0109-9
  7. Calvaruso, G., 10.1007/s00009-017-1019-2, Mediterr. J. Math. 14 (2017), 1–21. (2017) MR3707300DOI10.1007/s00009-017-1019-2
  8. Calvaruso, G., Zaeim, A., 10.1016/j.geomphys.2014.02.007, J. Geom. Phys. 80 (2014), 15–25. (2014) MR3188790DOI10.1016/j.geomphys.2014.02.007
  9. Fastenakels, J., Munteanu, M.I., Van Der Veken, J., Constant angle surfaces in the Heisenberg group, J. Acta Math. 27 (4) (2011), 747–756. (2011) MR2776411
  10. Gadea, P.M., Oubina, J.A., 10.1007/BF01320735, Monatsh. Math. 124 (1997), 17–34. (1997) MR1457209DOI10.1007/BF01320735
  11. Gadea, P.M., Oubina, J.A., 10.1007/s000130050403, Arch. Math. (Basel) 73 (1999), 311–320. (1999) MR1710084DOI10.1007/s000130050403
  12. Gil-Medrano, O., Hurtado, A., 10.1016/j.geomphys.2003.09.008, J. Geom. Phys. 51 (2004), 82–100. (2004) MR2078686DOI10.1016/j.geomphys.2003.09.008
  13. Gray, A., 10.1007/BF00151525, Geom. Dedicata 7 (1978), 259–280. (1978) Zbl0378.53018MR0505561DOI10.1007/BF00151525
  14. Nasehi, M., Aghasi, M., On the geometrical properties of hypercomplex four-dimensional Lorentzian Lie groups, to appear in Georgian Math. J. MR4069964
  15. Nasehi, M., Aghasi, M., 10.1016/j.geomphys.2018.06.008, J. Geom. Phys. 132 (2018), 230–238. (2018) MR3836779DOI10.1016/j.geomphys.2018.06.008
  16. Nasehi, M., Aghasi, M., 10.21136/CMJ.2018.0635-16, Czechoslovak Math. J. 68 (3) (2018), 723–740. (2018) MR3851887DOI10.21136/CMJ.2018.0635-16
  17. Nurowski, P., Randall, M., 10.1007/s12220-015-9592-8, J. Geom. Anal. 26 (2) (2016), 1280–1345. (2016) MR3472837DOI10.1007/s12220-015-9592-8
  18. Rahmani, N., Rahmani, S., 10.1016/0393-0440(94)90033-7, J. Geom. Phys. 13 (1994), 254–258. (1994) MR1269242DOI10.1016/0393-0440(94)90033-7
  19. Rahmani, N., Rahmani, S., 10.1007/s10711-005-9030-3, Geom. Dedicata 118 (2006), 133–140. (2006) MR2239452DOI10.1007/s10711-005-9030-3
  20. Rahmani, S., 10.1016/0393-0440(92)90033-W, J. Geom. Phys. 9 (3) (1992), 295–302, (French). (1992) MR1171140DOI10.1016/0393-0440(92)90033-W
  21. Tricerri, F., Vanhecke, L., Homogeneous structures on Riemannian manifolds, Cambridge University Press, 1983. (1983) MR0712664

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