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INVARIANT PSEUDO-SASAKIAN AND K-CONTACT STRUCTURES ON SEVEN-DIMENSIONAL NILPOTENT LIE GROUPS
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INVARIANT PSEUDO-SASAKIAN AND K-CONTACT STRUCTURES ON SEVEN-DIMENSIONAL NILPOTENT LIE GROUPS
Smolentsev Nikolai 1
Authors:
1.  Kemerovo State University

Russian Federation
Type:
Article
DOI:
https://doi.org/10.21603/2500-1418-2017-2-1-91-99
Pages:
from 91 to 99
Status:
Published
Received:
27.06.2017
Accepted:
27.06.2017
Published:
27.06.2017
Subject area:
GRNTI 29.01 Общие вопросы физики
GRNTI 27.01 Общие вопросы математики
GRNTI 31.01 Общие вопросы химии
GRNTI 34.01 Общие вопросы биологии
Language:
English
Keywords:
Nilmanifolds, seven-dimensional nilpotent Lie algebras, left-invariant contact structures, K-contact structures, Sasaki structures, pseudo-Kähler Lie groups, symplectic Lie groups
Abstract and keywords
Abstract (English):
This paper studies the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group A complete list of 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Riemannian Sasaki structures is found. A list of 25 classes of seven-dimensional nilpotent Lie groups admitting K-contact structures, but not pseudo-Riemannian Sasaki structures, is also presented. All the contact structures considered are central extensions of six-dimensional nilpotent symplectic Lie groups. Formulas that connect the geometric characteristics of six-dimensional nilpotent almost pseudo-Kähler Lie groups and seven-dimensional nilpotent contact Lie groups are established. As is known, for six-dimensional nilpotent pseudo-Kähler Lie groups the Ricci tensor is always zero. In contrast to the pseudo-Kӓhler case, it is shown that on contact seven-dimensional Lie algebras the Ricci tensor is nonzero even in directions of the contact distribution

Keywords:
Nilmanifolds, seven-dimensional nilpotent Lie algebras, left-invariant contact structures, K-contact structures, Sasaki structures, pseudo-Kähler Lie groups, symplectic Lie groups
Text
The left-invariant Kähler structure on a Lie group H is a triple (h, ω, J) consisting of a left-invariant Riemannian metric h, a left-invariant symplectic form ω, and an orthogonal left-invariant complex structure J, where h(X,Y) = ω(X,JY) for any left-invariant vector fields X and Y on H. Therefore, such a structure on the group H can be defined as a pair (ω, J), where ω is a symplectic form, and J is a complex structure compatible with ω, that is, such that ω(JX,JY) = ω(X,Y). If ω(X,JX) > 0, ∀ X # 0, then we obtain a Kӓhler metric, and if the positivity condition is not met, then h(X,Y) = ω(X,JY) is a pseudo-Riemannian metric, and then (h, ω, J) is called a pseudo-Kähler structure on the Lie group H. The left-invariance of these objects implies that the (pseudo) Kähler structure (h, ω, J) can be defined by the values of ω, J and h on the Lie algebra h of the Lie group H. Then (h, ω, J, h) is called a pseudo-Kähler Lie algebra. Conversely, if (h, J, h) is a Lie algebra endowed with a complex structure J orthogonal with respect to the pseudo-Riemannian metric h, then the equality ω(X,Y) = h(JX,Y) determines a (fundamental) 2-form ω which is closed if and only if J is parallel [10]. Classification of six-dimensional real nilpotent Lie algebras admitting invariant complex structures and estimation of the dimensions of moduli spaces of such structures is obtained in [11]. In [7], classification of symplectic structures on six-dimensional nilpotent Lie algebras is obtained. The condition of existence of left-invariant positive definite Kähler metric on the Lie group H imposes strong restrictions on the structure of its Lie algebra h. For example, Benson C. and Gordon C. [1] have shown that such a Lie algebra cannot be nilpotent except for the abelian case. Although nilpotent Lie groups and nilmanifolds (with the exception of the torus) do not admit left-invariant Kähler metrics, however, such manifolds may exist as left-invariant pseudo-Riemannian Kähler metrics. A complete list of six-dimensional pseudo-Kähler Lie algebras is given in [5]. A more complete study of the properties of the curvature of such pseudo-Kähler structures is carried out in [12]. In the odd-dimensional case, the analogues of symplectic structures are contact structures [2]. As is known [7], the contact Lie algebras g are central extensions of symplectic Lie algebras (h,ω) by means of a non-degenerate cocycle ω. In this case, a contact Lie algebra g admits Sasakian structure only if the Lie algebra (h,ω) admits a Kӓhler metric. Therefore, the question of the existence of Sasakian structures on seven-dimensional nilpotent Lie contact algebras g is reduced to the question of the existence of Kähler structures on six-dimensional nilpotent symplectic Lie algebras h = g/Z(g), where Z(g) is the center of the contact Lie algebra g. Classification of seven-dimensional nilpotent contact Lie algebras is obtained in [9]. In [4], examples are found of K-contact, but not Sasakian structures on seven-dimensional nilpotent contact Lie algebras.
References

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12. 12. Smolentsev N.K. Kanonicheskie psevdokelerovy metriki na shestimernykh nil’potentnykh gruppakh Li [Canonical pseudo-Kahler structures on six-dimensional nilpotent Lie groups]. Vestnik KemGU [Kemerovo State University Bulletin], 2011, no. 3/1 (47), pp. 155-168.

13. 13. Smolentsev N.K. Canonical almost pseudo-Kӓhler structures on six-dimensional nilpotent Lie groups. Differential Geometry, 2013, part 1, pp. 95-105. Available at: https://arxiv.org/abs/1310.5395

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