Torsion of SU(2)-structures and Ricci curvature in dimension 5

Abstract

In anlogy with the work of R. Bryant on the Ricci tensor of a G$_2$-structure, we study the intrinsic torsion of an SU$(2)$-structure on a 5-dimensional manifold deriving an explicit expression for the Ricci and the scalar curvature in terms of torsion forms and its derivative. As a consequence of this formula we prove that the $\alpha$-Einstein condition forces some special SU(2)-structures to be Sasaki-Einstein.