On the geometrical properties of hypercomplex four-dimensional Lie groups
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Mehri Nasehi
Abstract
In this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.
Acknowledgements
The authors are deeply grateful to the referee for the valuable comments and helpful suggestions.
References
[1] M. Aghasi and M. Nasehi, Some geometrical properties of a five-dimensional solvable Lie group, Differ. Geom. Dyn. Syst. 15 (2013), 1–12. Suche in Google Scholar
[2] M. Aghasi and M. Nasehi, On the geometrical properties of solvable Lie groups, Adv. Geom. 15 (2015), no. 4, 507–517. 10.1515/advgeom-2015-0025Suche in Google Scholar
[3] M. L. Barberis, Hypercomplex structures on four-dimensional Lie groups, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1043–1054. 10.1090/S0002-9939-97-03611-3Suche in Google Scholar
[4] M. L. Barberis, Hyper-Kähler metrics conformal to left invariant metrics on four-dimensional Lie groups, Math. Phys. Anal. Geom. 6 (2003), no. 1, 1–8. 10.1023/A:1022448007111Suche in Google Scholar
[5] G. Calvaruso, Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups, J. Geom. Phys. 61 (2011), no. 2, 498–515. 10.1016/j.geomphys.2010.11.001Suche in Google Scholar
[6] G. Calvaruso, Harmonicity of vector fields on four-dimensional generalized symmetric spaces, Cent. Eur. J. Math. 10 (2012), no. 2, 411–425. 10.2478/s11533-011-0109-9Suche in Google Scholar
[7] G. Calvaruso and J. Van der Veken, Totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups, Results Math. 64 (2013), no. 1–2, 135–153. 10.1007/s00025-012-0304-4Suche in Google Scholar
[8] G. Calvaruso and A. Zaeim, A complete classification of Ricci and Yamabe solitons of non-reductive homogeneous 4-spaces, J. Geom. Phys. 80 (2014), 15–25. 10.1016/j.geomphys.2014.02.007Suche in Google Scholar
[9] E. Calviño Louzao, J. Seoane-Bascoy, M. E. Vázquez-Abal and R. Vázquez-Lorenzo, Three-dimensional homogeneous Lorentzian Yamabe solitons, Abh. Math. Semin. Univ. Hambg. 82 (2012), no. 2, 193–203. 10.1007/s12188-012-0072-9Suche in Google Scholar
[10] B. De Leo and J. Van der Veken, Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces, Geom. Dedicata 159 (2012), 373–387. 10.1007/s10711-011-9665-1Suche in Google Scholar
[11] G. W. Gibbons, G. Papadopoulos and K. S. Stelle, HKT and OKT geometries on soliton black hole moduli spaces, Nuclear Phys. B 508 (1997), no. 3, 623–658. 10.1016/S0550-3213(97)00599-3Suche in Google Scholar
[12] A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata 7 (1978), no. 3, 259–280. 10.1007/BF00151525Suche in Google Scholar
[13] H. R. S. Moghaddam, Randers metrics of Berwald type on four-dimensional hypercomplex Lie groups, J. Phys. A: Math. Theor. 42 (2009), no. 9, Article ID 095212. 10.1088/1751-8113/42/9/095212Suche in Google Scholar
[14] D. Perrone, Almost contact metric manifolds whose Reeb vector field is a harmonic section, Acta Math. Hungar. 138 (2013), no. 1–2, 102–126. 10.1007/s10474-012-0228-1Suche in Google Scholar
[15] Y. S. Poon, Examples of hyper-Kähler connections with torsion, Quaternionic Structures in Mathematics and Physics (Rome 1999), Universitá Studi Roma “La Sapienza”, Rome (1999), 321–327. 10.1142/9789812810038_0020Suche in Google Scholar
[16] H. R. Salimi Moghaddam, On some hypercomplex 4-dimensional Lie groups of constant scalar curvature, Int. J. Geom. Methods Mod. Phys. 6 (2009), no. 4, 619–624. 10.1142/S0219887809003710Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials
- On generalized α-ψ-Geraghty contractions on b-metric spaces
- New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods
- A Tauberian theorem for the generalized Nörlund summability method
- A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities
- The Robin function and conformal welding – A new proof of the existence
- Effects of the initial moment and several delays perturbations in the variation formulas for a solution of a functional differential equation with the continuous initial condition
- The well-posedness of a nonlocal multipoint problem for a differential operator equation of second order
- Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations
- On an approximate solution of a class of surface singular integral equations of the first kind
- On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
- On the geometrical properties of hypercomplex four-dimensional Lie groups
- On sets of singular rotations for translation invariant differentiation bases formed by intervals
- Certain commutativity criteria for rings with involution involving generalized derivations
- The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds
- Ripplet transform and its extension to Boehmians
- Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces
Artikel in diesem Heft
- Frontmatter
- An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials
- On generalized α-ψ-Geraghty contractions on b-metric spaces
- New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods
- A Tauberian theorem for the generalized Nörlund summability method
- A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities
- The Robin function and conformal welding – A new proof of the existence
- Effects of the initial moment and several delays perturbations in the variation formulas for a solution of a functional differential equation with the continuous initial condition
- The well-posedness of a nonlocal multipoint problem for a differential operator equation of second order
- Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations
- On an approximate solution of a class of surface singular integral equations of the first kind
- On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
- On the geometrical properties of hypercomplex four-dimensional Lie groups
- On sets of singular rotations for translation invariant differentiation bases formed by intervals
- Certain commutativity criteria for rings with involution involving generalized derivations
- The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds
- Ripplet transform and its extension to Boehmians
- Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces
