Foliations with Transversal Quaternionic Structures

Abstract

We consider manifolds equipped with a foliation $\mathcal{F}$ of codimension $4q$, and an almost quaternionic structure $Q$ on the transversal bundle of $\mathcal{F}$. After discussing conditions of projectability and integrability of $Q$, we study the transversal twistor space $Z\mathcal{F}$ which, by definition, consists of the $Q$-compatible almost complex structures. We show that $Z\mathcal{F}$ can be endowed with a lifted foliation $\hat{\mathcal{F}}$ and two natural almost complex structures $J_1$, $J_2$ on the transversal bundle of $\hat{\mathcal{F}}$. 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).

LaTex, 36 pages