Quaternionic and para-quaternionic CR manifolds
by
Dmitri Alekseevsky
Hull University, Department of Mathematics
Coauthors: Y. Kamishima

Notions of quaternionic and para-quaternionic CR structures on a (4n+3) -dimension manifold M are defined as a triple of 1-forms w = (w₁, w₂, w₃) which satisfy some conditions. We associate with such a structure a pseudo-Riemannian Einstein metric g on M and a Lie algebra a of its Killing fieleds , isomorphic to sp(1) in quaternionic case and sp(1, R) in para-quaternionic case. If the metric g is positively defined, then a quaternionic CR structure is equivalent to a Sasakian 3-structure. We give examples of homogeneous manifolds with invariant quaternionic and para-quaternionic CR structure and describe a reduction method, which allows to construct non-homogeneous quaternionic and para-quaternionic CR structures starting from a manifold with such a structure which has a symmetry Lie group. It is shown also that a cone over a manifold M with (para)-quaternionic CR structure carries a (para)-hyperKaehler structure and the quotient M/A of M by the Lie group A of isometries, generated by a, carries a (para)-quaternionic Kaehler structure (under assumption that the group A is defined and acts properly on M.)

Date received: April 26, 2003