The identification of conformal hypercomplex and quaternionic manifolds

Details The identification of conformal hypercomplex and quaternionic manifolds The identification of conformal hypercomplex and quaternionic manifolds

2005-12-04

arxiv.org/abs/math/0512084v1

Int.J.Geom.Meth.Mod.Phys. 3 (2006) 913-932

Mathematics

Differential Geometry

22 pages, 2 figures, Contribution to the proceedings volume for the Conference "Symmetry in Geometry and Physics" in honour of

Scientific paper

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 `\xi transformations', relating connections on quaternionic manifolds, and a new type `\hat\xi transformations' relating complex structures on conformal hypercomplex manifolds. In this map, the subclass of conformal hyper-Kaehler manifolds is mapped to quaternionic-Kaehler manifolds.

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