Para-Hermitian and Para-Quaternionic manifolds

A set of canonical parahermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraK"ahler manifolds is parallel with respect to the canonical connection. Salamon’s twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on $So{l}_1^4$. A locally conformally flat hyper-paraK"ahler (hypersymplectic) structure with parallel Lee form on Kodaira-Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on $So{l}_1^4$ is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraK"ahler is constructed.