On the Geometry of Higher Dimensional Heisenberg Groups

  • Mehri NasehiEmail author


In this paper, we first completely determine all left-invariant generalized Ricci solitons on the Heisenberg group \(H_{2n+1}\) equipped with any left-invariant Riemannian and Lorentzian metric that this Lie group admits. Then, we explicitly calculate the energy of an arbitrary left-invariant vector field V on these spaces and in the Lorentzian cases we determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also obtain all of the descriptions of their homogeneous Riemannian and Lorentzian structures and explicitly distinguish their types. Finally, we investigate parallel hypersurfaces of these spaces and show that these spaces never admit any totally geodesic hypersurface. The existence of algebraic Ricci solitons and the non-existence of left-invariant Ricci solitons and Yamabe solitons on these spaces in both Riemannian and Lorentzian cases is proved. Also, different behaviors regarding the existence of harmonic maps, critical points for the energy functional restricted to vector fields and some equations in Riemannian and Lorentzian cases are found.


Heisenberg groups Left-invariant generalized Ricci solitons Harmonicity of invariant vector fields Parallel and totally geodesic hypersurfaces Homogeneous structures 

Mathematics Subject Classification

53C30 53C50 



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Basic SciencesUniversity of Shahreza ShahrezaIran

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