On the geometry of some solvable extensions of the Heisenberg group

Mehri Nasehi; Mansour Aghasi

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 3, page 723-740
  • ISSN: 0011-4642

Abstract

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In this paper we first classify left-invariant generalized Ricci solitons on some solvable extensions of the Heisenberg group in both Riemannian and Lorentzian cases. Then we obtain the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and obtain all vector fields which are critical points for the energy functional restricted to vector fields of the same length. Furthermore, we determine all homogeneous Lorentzian structures and their types on these spaces and give a complete and explicit description of all parallel and totally geodesic hypersurfaces of these spaces. The non-existence of harmonic maps in the non-abelian case is proved and it is shown that the existence of Einstein, Einstein-like metrics and some equations in the Riemannian case can not be extended to their Lorentzian analogues.

How to cite

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Nasehi, Mehri, and Aghasi, Mansour. "On the geometry of some solvable extensions of the Heisenberg group." Czechoslovak Mathematical Journal 68.3 (2018): 723-740. <http://eudml.org/doc/294313>.

@article{Nasehi2018,
abstract = {In this paper we first classify left-invariant generalized Ricci solitons on some solvable extensions of the Heisenberg group in both Riemannian and Lorentzian cases. Then we obtain the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also calculate the energy of an arbitrary left-invariant vector field $X$ on these spaces and obtain all vector fields which are critical points for the energy functional restricted to vector fields of the same length. Furthermore, we determine all homogeneous Lorentzian structures and their types on these spaces and give a complete and explicit description of all parallel and totally geodesic hypersurfaces of these spaces. The non-existence of harmonic maps in the non-abelian case is proved and it is shown that the existence of Einstein, Einstein-like metrics and some equations in the Riemannian case can not be extended to their Lorentzian analogues.},
author = {Nasehi, Mehri, Aghasi, Mansour},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized Ricci soliton; harmonicity of vector field; homogeneous Lorentzian structure; parallel hypersurfaces},
language = {eng},
number = {3},
pages = {723-740},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the geometry of some solvable extensions of the Heisenberg group},
url = {http://eudml.org/doc/294313},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Nasehi, Mehri
AU - Aghasi, Mansour
TI - On the geometry of some solvable extensions of the Heisenberg group
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 723
EP - 740
AB - In this paper we first classify left-invariant generalized Ricci solitons on some solvable extensions of the Heisenberg group in both Riemannian and Lorentzian cases. Then we obtain the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also calculate the energy of an arbitrary left-invariant vector field $X$ on these spaces and obtain all vector fields which are critical points for the energy functional restricted to vector fields of the same length. Furthermore, we determine all homogeneous Lorentzian structures and their types on these spaces and give a complete and explicit description of all parallel and totally geodesic hypersurfaces of these spaces. The non-existence of harmonic maps in the non-abelian case is proved and it is shown that the existence of Einstein, Einstein-like metrics and some equations in the Riemannian case can not be extended to their Lorentzian analogues.
LA - eng
KW - generalized Ricci soliton; harmonicity of vector field; homogeneous Lorentzian structure; parallel hypersurfaces
UR - http://eudml.org/doc/294313
ER -

References

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