A class of Sasakian 5-manifolds

Abstract

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H 2n+1 . Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H 5 or a semidirect product R (H 3 × R). In particular, the compact quotient is an S 1 -bundle over a four-dimensional Kähler solvmanifold.

Almost Contact Metric Structures on 5-Dimensional Nilpotent Lie Algebras

“…By considering Φ as the fundamental two-form of an almost contact metric structure (φ, ξ, η, g), from the condition φ 2 = −I + η ⊗ ξ, we get b 2 15 = 1 and b 45 = 0. We get b 15 = b 24 = 0 for X = e 5 , Y = e 2 , Z = e 3 and X = e 1 , Y = e 3 , Z = e 4 respectively in the equation (2).…”

“…In literature, some certain classes of such structures are studied. In [2], some general results on 5-dimensional Sasakian Lie algebras were stated, and it was proved that an odd dimensional nilpotent Lie group with a left invariant Sasakian structure is isomorphic to the real Heisenberg group. In addition, a classification of five-dimensional Sasakian Lie algebras were obtained.…”

“…Note that this structure is given in [2] as a Sasakian structure because of the coefficient 2 in the defining relation of a Sasakian structure.…”

Five-dimensional K-contact Lie algebras

“…While the converse is also true in dimension three, in dimension greater or equal to five a K -contact manifold needs not to be Sasakian. In [2], A. Andrada, L. Vezzoni and A. Fino obtained some general results on Sasakian Lie algebras and their full classification in dimension five.…”

“…This study is the purpose of the present paper. We start from the general result on the center of a contact Lie algebra proved in [2]: a contact Lie algebra either has trivial center, or its center is one-dimensional and spanned by the characteristic vector field.…”