Para-Hermitian and Para-Quaternionic manifolds

TL;DR

This paper develops canonical connections for para-Hermitian manifolds, extends classical theorems to the para setting, and constructs explicit examples of hyper-paraK"ahler and anti-self-dual metrics on specific surfaces.

Contribution

It introduces canonical parahermitian connections, generalizes Goldberg-Sachs theorem to paraHermitian manifolds, and constructs explicit hyper-paraK"ahler and anti-self-dual metrics on Kodaira-Thurston and Inoe surfaces.

Findings

Nijenhuis tensor of Nearly paraK"ahler manifolds is parallel w.r.t. canonical connection

Constructed hyper-paracomplex structures on specific surfaces

Presented anti-self-dual neutral metrics with novel properties

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 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.