On some solvable extensions of the Heisenberg group II

In this paper we consider two families of four or five-dimensional solvable Lie groups which are extensions of the Heisenberg group. We first calculate the energy of an arbitrary left-invariant vector field X on these spaces. Then we obtain the exact form of all harmonic maps on these spaces and prove that if these spaces admit a non-trivial harmonic map, then any harmonic map is a critical point for the energy functional restricted to vector fields of the same length. Later on, we investigate the existence ofleft-invariant Yamabe solitons and explicitly describe all geodesic vectors on these spaces.

Keywords:

Solvable Lie groups , Heisenberg group , Harmonic maps , Yamabe solitons , geodesic vectors.