Para-Hermitian and Para-Quaternionic manifolds

Details Para-Hermitian and Para-Quaternionic manifolds Para-Hermitian and Para-Quaternionic manifolds

2003-10-26

arxiv.org/abs/math/0310415v4

Differ.Geom.Appl. 23 (2005) 205-234

Mathematics

Differential Geometry

LaTeX2e, 27 pages, final version, to appear in Diff. Geom. Appl

Scientific paper

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 $Sol^4_1$. 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 $Sol^4_1$ is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraK\"ahler is constructed.

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