International Journal of Geometric Methods in Modern Physics vol:3 issue:5-6 pages:913-932
We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. "conformal hypercomplex" manifolds) and quaternionic manifolds of one dimension less. This map is related to a method for constructing supergravity theories using superconformal techniques. An explicit relation between the structure of these manifolds is presented, including curvatures and symmetries. An important role is played by "xi transformations," relating connections on quaternionic manifolds, and a new type "(xi) over cap transformations" relating complex structures on conformal hypercomplex manifolds. In this map, the subclass of conformal hyper-Kahler manifolds is mapped to quaternionic-Kahler manifolds.