Left-invariant almost para-Hermitian structures on some six-dimensional nilpotent Lie groups

As is well known, there are $34$ classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only $26$ classes admit left-invariant symplectic structures and only $18$ classes admit left-invariant complex structures. There exist five six-dimensional nilpotent Lie groups $G$, which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo-Kählerian, nor Hermitian. It is the Lie groups that are studied in this work. The aim of the paper is to define new left-invariant geometric structures on the Lie groups. If the left-invariant $2$-form $\omega$ on such a Lie group is closed, then it is degenerate. Weakening the closedness requirement for left-invariant $2$-forms $\omega$, stable $2$-forms $\omega$ are obtained. Their exterior differential $d\omega$ is also stable in Hitchin sense. Therefore, the pair $(\omega, d\omega)$ defines either an almost Hermitian or almost para-Hermitian structure on the group $G$. The corresponding pseudo-Riemannian metrics are Einstein for four of the five Lie groups under consideration. This gives new examples of multiparameter families of left-invariant Einstein pseudo-Riemannian metrics on six-dimensional nilmanifolds. On each of the Lie groups under consideration, compatible and normalized pairs of left-invariant forms $(\omega,\rho)$, where $\rho=d\omega$, are obtained. They define semi-flat structures. The Hitchin flow on $G\times I$ is studied to construct a pseudo-Riemannian metric on $G\times I$ with a holonomy group from $G_2^*$ and it is shown that there is nots solution in this class of left-invariant half-plane structures $(\omega,\rho)$. For structures $(\omega,\rho)$, only the $3$-form closure property $\varphi=\omega \wedge dt+d\omega$ on $G\times I$ holds.