Left-invariant almost para-Hermitian structures on some sixdimensional nilpotent Lie groups | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2019. № 58. DOI: 10.17223/19988621/58/4

Left-invariant almost para-Hermitian structures on some sixdimensional 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-Kahlerian, 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 ю on such a Lie group is closed, then it is degenerate. Weakening the closedness requirement for left-invariant 2-forms ю, stable 2-forms ю are obtained. Their exterior differential dw is also stable in Hitchin sense. Therefore, the pair (ю, di») 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 (ю, p), where p = dm, are obtained. They define semi-flat structures. The Hitchin flow on G x / is studied to construct a pseudo-Riemannian metric on G x / with a holonomy group from G2^* and it is shown that there is nots solution in this class of left-invariant half-plane structures (ю, p). For structures (ю, p), only the 3-form closure property Ф = ffiAdt + dw on Gx/ holds. AMS Mathematical Subject Classification: 53C15, 53C30, 53C25, 22E25

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Keywords

нильмногообразия, шестимерные нильпотентные алгебры Ли, левоинвариантные пара-комплексные структуры, эйнштейновы многообразия, полуплоские структуры, nilmanifolds, six-dimensional nilpotent Lie algebras, left-invariant para-complex structures, Einstein manifolds, half-flat structures

Authors

Name	Organization	E-mail
Smolentsev Nikolay K.	Kemerovo State University	smolennk@mail.ru

Всего: 1

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