A class of Sasakian 5-manifolds

TL;DR

This paper classifies 5-dimensional Sasakian Lie algebras, showing that nilpotent cases are isomorphic to the Heisenberg group and identifying structures that admit compact quotients, with implications for Sasakian geometry.

Contribution

It provides a classification of 5-dimensional Sasakian Lie algebras and characterizes those with Sasakian $ ext{α}$-Einstein structures, including the structure of compact quotients.

Findings

Nilpotent Sasakian Lie groups are isomorphic to the Heisenberg group.

Classified 5-dimensional Sasakian Lie algebras and identified which admit Sasakian $ ext{α}$-Einstein structures.

Described the structure of 5-dimensional solvable Lie groups with compact quotients.

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 $\R \ltimes (H_3 \times \R)$. In particular, the compact quotient is an $S^1$-bundle over a 4-dimensional K\"ahler solvmanifold.