The identification of conformal hypercomplex and quaternionic manifolds
arXiv:math/0512084 · doi:10.1142/S0219887806001521
Abstract
We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. `conformal hypercomplex' manifolds) and quaternionic manifolds of 1 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 `ξtransformations', relating connections on quaternionic manifolds, and a new type `\hatξtransformations' relating complex structures on conformal hypercomplex manifolds. In this map, the subclass of conformal hyper-Kaehler manifolds is mapped to quaternionic-Kaehler manifolds.
22 pages, 2 figures, Contribution to the proceedings volume for the Conference "Symmetry in Geometry and Physics" in honour of Dmitri Alekseevsky, September 2005