Twistor space of a generalized quaternionic manifold

TL;DR

This paper surveys twistor theory for various quaternionic manifolds and provides an integrability criterion for the generalized complex structure on their twistor spaces, with examples in generalized hyperk"ahler manifolds.

Contribution

It introduces an integrability criterion for the generalized complex structure on twistor spaces of quaternionic manifolds and explores examples using the generalized Bismut connection.

Findings

Established integrability conditions for the generalized complex structure.

Provided examples in the context of generalized hyperk"ahler manifolds.

Connected twistor theory with generalized Bismut connection applications.

Abstract

We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was iniated by Pantilie, who shows that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space with an almost generalized complex structure. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperk\"ahler manifolds using the generalized Bismut connection, introduced by Gualtieri.