Bianchi type A hyper-symplectic and hyper-K\"ahler metrics in 4D

TL;DR

This paper introduces a straightforward method to construct hyper-Kaehler and hyper-symplectic metrics in four dimensions using Bianchi type A groups, revealing a connection between these geometric structures.

Contribution

It provides an explicit construction method for hyper-Kaehler and hyper-symplectic metrics in 4D based on Bianchi type A groups, highlighting a correspondence between these structures.

Findings

Explicit construction of 4D hyper-Kaehler metrics

Explicit construction of 4D hyper-symplectic metrics

Identification of a correspondence between hyper-Kaehler and hyper-symplectic structures

Abstract

We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper-symplectic structures in dimension four.