Para-quaternionic reduction

Original · EN

The pseudo-Riemannian manifold M=(M⁴ⁿ,g), n ≥ 2 is para-quaternionic K ahler if hol(M) ⊂ sp(n,) ⊕ sp(1,). If hol(M) ⊂ sp(n,), than the manifold M is called para-hyperK ahler. The other possible definitions of these manifolds use certain parallel para-quaternionic structures in (TM), similarly to the quaternionic case. In order to relate these different definitions we study para-quaternionic algebras in details. We describe the reduction method for the para-quaternionic K ahler and para-hyperK ahler manifolds and give some examples. The decomposition of a curvature tensor of the para-quaternionic type is also described.