On the geometrical properties of Heisenberg groups

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.