arXiv subjects
Publications and source records attributed to Marina Tchomakova.
We consider certain fiber bundles over a paraquaternionic contact manifolds, called twistor and reflector spaces, and show that these carry an intrinsic geometric structure that is always integrable.
A tensor invariant is defined on a paraquaternionic contact manifold in terms of the curvature and torsion of the canonical paraquaternionic connection involving derivatives up to third order of the contact form. This tensor, called paraquaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry, the Chern-Moser tensor in CR geometry, the para contact curvature in para CR geometry and to the quaternionic contact conformal curvature in quaternionic contact geometry. It is shown that a paraquaternionic contact manifold is locally paraquaternionic contact conformal to the standard flat paraquaternionic contact structure on the paraquaternionic Heisenberg group, or equivalently, to the standard para 3-Sasakian structure on the paraquaternionic pseudo-sphere iff the paraquaternionic contact conformal curvature vanishes.
We introduce the notion of paraquaternionic contact structures (pqc structures), which turns out to be a generalization of the para 3-Sasakian geometry. We derive a distinguished linear connection preserving the pqc structure. Its torsion tensor is expressed explicitly in terms of the structure tensors and the structure equations of a pqc manifold are presented. We define pqc-Einstein manifolds and show that para 3-Sasakian spaces are precisely pqc manifolds, which are pqc-Einstein. Furthermore, we introduce the paraquaternionic Heisenberg qroup and show that it is the flat model of the pqc geometry.