Foliations with Transversal Quaternionic Structures

TL;DR

This paper explores manifolds with foliations and transversal quaternionic structures, analyzing the properties of the associated transversal twistor space and the conditions for complex structure integrability.

Contribution

It introduces conditions for projectability and integrability of almost complex structures on the transversal twistor space of quaternionic foliations.

Findings

The transversal twistor space admits a lifted foliation with natural almost complex structures.

Conditions for the projectability of these structures are established.

The integrability of one of the almost complex structures is characterized, while the other is never integrable.

Abstract

We consider manifolds equipped with a foliation $\cal F$ of codimension $4q$, and an almost quaternionic structure $Q$ on the transversal bundle of ${\cal F}$. After discussing conditions of projectability and integrability of $Q$, we study the transversal twistor space $Z{\cal F}$ which, by definition, consists of the $Q$-compatible almost complex structures. We show that $Z{\cal F}$ can be endowed with a lifted foliation ${\hat {\cal F}}$ and two natural almost complex structures $J_1$, $J_2$ on the transversal bundle of $\hat{\cal F}$. We establish the conditions which ensure the projectability of $J_1$ and $J_2$, and the integrability of $J_{1}$ ($J_{2}$ is never integrable).