On the geometrical properties of hypercomplex four-dimensional Lie groups

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.

On the geometry of some solvable extensions of the Heisenberg group

“…, such that the components B ijk can be obtained by the relation ∇ ei e j = k ε j B ijk e k (for more details see [15]). Thus we can obtain the following result.…”