Invariant $\mathit{f}$-structures on the four-dimensional oscillator group

In this paper, we investigate the four-dimensional oscillator group from the point of view of the generalised Hermitian geometry. This solvable Lie group is a semi-direct product of the classical three-dimensional Heisenberg group by a real line. Using the corresponding Lie algebra, we construct and study six basic left-invariant metric $\mathit{f}$-structures of rank $2$ on the oscillator group. As a result, it gives the opportunity to present new examples of left-invariant nearly Kāhler, generalised nearly Kāhler and Hermitian $\mathit{f}$-structures on solvable Lie groups.