Foliations with Transversal Quaternionic Structures

Details Foliations with Transversal Quaternionic Structures Foliations with Transversal Quaternionic Structures

2000-01-11

arxiv.org/abs/math/0001059v1

Mathematics

Differential Geometry

LaTex, 36 pages

Scientific paper

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).

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