Invariant pseudo-Sasakian and $K$-contact structures on seven-dimensional nilpotent Lie groups

Abstract

We study the question of 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. We find a complete list of the 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Sasaki structure. We also present a list of 25 classes of seven-dimensional nilpotent Lie groups admitting a $K$-contact structure, but not the pseudo-Sasaki structure. All the contact structures considered are central extensions of six-dimensional nilpotent symplectic Lie groups and are established formulas that connect the geometrical characteristics of the six-dimensional nilpotent almost pseudo-K\"{a}hler Lie groups and seven-dimensional nilpotent contact Lie groups. It is known that for the six-dimensional nilpotent pseudo-K\"{a}hler Lie groups the Ricci tensor is always zero. Unlike the pseudo-K\"{a}hlerian case, it is shown that on contact seven-dimensional algebras the Ricci tensor is nonzero even in directions of the contact distribution.