Para-Hermitian and Para-Quaternionic manifolds

Abstract

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 paraKähler 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-paraKähler (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-paraKähler is constructed.

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

Cited by in corpus (9)

• A Unique Connection for Born Geometry
• Algebroid Structures on Para-Hermitian Manifolds
• Pseudo-hyperkahler Geometry and Generalized Kahler Geometry
• Hyper-ParaHermitian manifolds with torsion
• Compact complex surfaces with geometric structures related to split quaternions
• Geometry of Paraquaternionic Kahler manifolds with torsion
• Geometric structures on Lie groups with flat bi-invariant metric
• Harmonic maps and para-Sasakian geometry
• Born Geometry in a Nutshell