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MATHEMATICS
Left-invariant almost para-Hermitian structures on some six-dimensional nilpotent Lie groups
N. K. Smolentsev Fundamental Mathematics department of Kemerovo State University, Kemerovo, Russian Federation
Abstract:
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.
Keywords:
nilmanifolds, six-dimensional nilpotent Lie algebras, left-invariant para-complex structures, Einstein manifolds, half-flat structures.
Received: 01.11.2018
Citation:
N. K. Smolentsev, “Left-invariant almost para-Hermitian structures on some six-dimensional nilpotent Lie groups”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2019, no. 58, 41–55
Linking options:
https://www.mathnet.ru/eng/vtgu698 https://www.mathnet.ru/eng/vtgu/y2019/i58/p41
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Abstract page: | 128 | Full-text PDF : | 59 | References: | 21 |
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