On a class of Sasakian 5-manifolds

ADRIÁN ANDRADA, ANNA FINO, LUIGI VEZZONI

Marburg

Workshop; Dirac operators and special geometries; 2009

Philipps-Universität Marburg

In this joint work with A. Fino and L. Vezzoni 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 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 $mathbb{R} \ltimes (H_3 \times \mathbb{R})$. In particular, the compact quotient is an S^1-bundle over a 4-dimensional Kähler solvmanifold.