Communications in mathematical physics vol:262 issue:2 pages:411-457
We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by 'xi-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kahler manifolds is mapped to quaternionic-Kahler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.