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 5 and determine which of them carry a Sasakian $\alpha$-Einstein structure. We show that a 5-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 $\mathbb{R}⋉(H_3\times \mathbb{R})$. In particular, the compact quotient is an $S^1$-bundle over a 4-dimensional Kähler solvmanifold.

The title has been shortened; references added or updated; typos corrected. Formula (4) corrected. Some changes in the introduction. To appear in Transformation Groups

References in corpus (1)

• Contact 5-manifolds with SU(2)-structure