# Handbook of Pseudo-Riemannian Geometry and Supersymmetry

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#### DOI: 10.4171/079-1/14

Twistor and reflector spaces of almost para-quaternionic manifolds

Stefan Ivanov[1], Ivan Minchev[2] and Simeon Zamkovoy[3]

(1) University of Sofia, Bulgaria
(2) University of Sofia, Bulgaria
(3) University of Sofia, Bulgaria

We investigate the integrability of almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of an arbitrary para-quaternionic connection. We show that if there exists an integrable structure then it is independent on the para-quaternionic connection. In dimension four, we express the ant-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.