Nearly pseudo-K\"ahler and nearly para-K\"ahler six-manifolds

TL;DR

This paper investigates six-dimensional nearly (para-)K"ahler manifolds with pseudo-Riemannian metrics, deriving differential systems, analyzing automorphisms, and establishing existence and uniqueness of structures on specific Lie groups.

Contribution

It introduces an analogue of the exterior differential system for nearly (para-)K"ahler manifolds and proves existence and uniqueness of such structures on certain Lie groups.

Findings

Derived the differential system characterising nearly (para-)K"ahler manifolds.

Analyzed automorphism groups of these structures.

Proved existence and uniqueness on Lie groups G×G with G three-dimensional and simple.

Abstract

The subject of this paper is six-dimensional nearly (para-)K\"ahler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly K\"ahler manifold and prove applications to the automorphism group of a nearly (para-)K\"ahler structure. Secondly, we prove existence and uniqueness results for left-invariant nearly (para-)K\"ahler structures on Lie groups $G \times G$ where $G$ is three-dimensional and simple.