Abstract
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
Highlights
A left-invariant Kahler structure on a Lie group H is a triple (h, ω, J), consisting of a leftinvariant Riemannian metric h, a left-invariant symplectic form ω and an orthogonal leftinvariant complex structure J, where h(X, Y ) = ω(X, JY ) for any left-invariant vector fields X and Y on H
If (h, J, h) is a Lie algebra endowed with a complex structure J, orthogonal with respect to the pseudo-Riemannian metric h, the equality ω(X, Y ) = h(JX, Y ) determines the two-form ω, which is closed if and only if the J is parallel [9]
We show that on seven-dimensional nilpotent Lie algebras the invariant contact Sasakian structures exist only in the case of the Heisenberg algebra, and pseudo-Sasakian there are only 22 classes of central extensions of six-dimensional pseudo Kahler nilpotent Lie algebras
Summary
A left-invariant Kahler structure on a Lie group H is a triple (h, ω, J), consisting of a leftinvariant Riemannian metric h, a left-invariant symplectic form ω and an orthogonal leftinvariant complex structure J, where h(X, Y ) = ω(X, JY ) for any left-invariant vector fields X and Y on H. We show that on seven-dimensional nilpotent Lie algebras the invariant contact Sasakian structures exist only in the case of the Heisenberg algebra, and pseudo-Sasakian (i.e., having a pseudo-Riemannian metric tensor) there are only 22 classes of central extensions of six-dimensional pseudo Kahler nilpotent Lie algebras. We obtain the list of 25 classes of seven-dimensional nilpotent contact Lie algebras that allow K-contact structure, but do not allow pseudo-Sasakian structures
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