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On the geometrical properties of Heisenberg groups. (English) Zbl 1463.53077
In the present paper, the geometry of the \(3\)-dimensional Heisenberg group \(H_3\) is studied. Three families of Lorentzian metrics and a family of Riemannian metrics on \(H_3\) are considered. Generalized Ricci solitons among these metrics are determined. Algebraic Ricci solitons among these metrics are determined and it is shown that none of them are invariant Yamabe solitons or invariant Ricci solitons. The homogeneous structures of the above metrics are also determined with their type. The harmonicity properties of these spaces are investigated. The energy of an invariant vector field is calculated and it is shown that for some metrics the vector field is a harmonic map. Invariant unit time-like vector fields which are spatially harmonic are determined. These results were already generalized by the author to the general dimension \(2n+1\) in [the author, Mediterr. J. Math. 16, No. 2, Paper No. 29, 17 p. (2019; Zbl 1420.53077)]. Related results on Lorentzian solitons on nilpotent Lie groups (mostly in dimension 5) with a general survey can be found in [T. H. Wears, Math. Nachr. 290, No. 8–9, 1381–1405 (2017; Zbl 1378.53061)].
MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
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